SELINA Solutions for Class 9 Maths Chapter 10 - Isosceles Triangle

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Chapter 10 - Isosceles Triangle Exercise Ex. 10(A)

Question 1

In the figure alongside,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA = 48o and

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD = 18o.

Show that BC = CD.

Solution 1

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBAC + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = 1800

480 + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = 1800

But Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC [AB = AC]

2Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = 1800 - 480

2Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = 1320

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = 660 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB ……(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB = 660

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCB = 660

180 + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCB = 660

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCB = 480 ………(ii)

Now, In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC = 660 [From (i), Since Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCB = 480 [From (ii)]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBDC = 1800 - 480 - 660

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBDC = 660

Since Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBDC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC

Therefore, BC = CD

Equal angles have equal sides opposite to them.

Question 2

Calculate:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADC

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBAC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 2

Given: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACE = 1300; AD = BD = CD

Proof:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 3

In the following figure, AB = AC; BC = CD and DE is parallel to BC. Calculate:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCDE

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCE

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 4

Calculate x:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 4

(i) Let the triangle be ABC and the altitude be AD.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii) Let triangle be ABC and altitude be AD.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 5

In the figure, given below, AB = AC. Prove that: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBOC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Let Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABO =Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleOBC = x and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACO = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleOCB = y

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 6

In the figure given below, LM = LN; angle PLN = 110o. Calculate:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleLMN

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleMLN

Solution 6

Given: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i) We know that the sum of the measure of all the angles of a quadrilateral is 360o.

In quad. PQNL,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 7

An isosceles triangle ABC has AC = BC. CD bisects AB at D and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAB = 55o.

Find: (i) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDCB (ii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBD.

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 8

Find x:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 8

Let us name the figure as following:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

For x:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 9

In the triangle ABC, BD bisects angle B and is perpendicular to AC. If the lengths of the sides of the triangle are expressed in terms of x and y as shown, find the values of x and y.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore,

AD=DC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

and AB = BC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Substituting the value of x from (i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Putting y = 3 in (i)

x = 3 + 1

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Trianglex = 4

Question 10

In the given figure; AE // BD, AC // ED and AB = AC. Find Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Trianglea, Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangleb and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Trianglec.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 10

Let P and Q be the points as shown below:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Given: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 11

In the following figure; AC = CD, AD = BD and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC = 58o.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Find angle CAB.

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 12

In the figure of q. no. 11 given above, if AC = AD = CD = BD; find angle ABC.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 12

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 13

In triangle ABC; AB = AC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA : Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB = 8 : 5; find angle A.

Solution 13

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Let Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Given: AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle [Angles opp. to equal sides are equal]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 14

In triangle ABC; Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA = 60o, Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC = 40o, and bisector of angle ABC meets side AC at point P. Show that BP = CP.

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

BP is the bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 15

In triangle ABC; angle ABC = 90o and P is a point on AC such that Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePCB. Show that: PA = PB.

Solution 15

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Let Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePCB = x

In the right angled triangle ABC,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

and

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore in the triangle ABP;

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Hence,

PA = PB [sides opp. to equal angles are equal]

Question 16

ABC is an equilateral triangle. Its side BC is produced upto point E such that C is mid-point of BE. Calculate the measure of angles ACE and AEC.

Solution 16

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 

Question 17

In triangle ABC, D is a point in AB such that AC = CD = DB. If B = 28°, find the angle ACD.

Solution 17

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Question 18

In the given figure, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. Find angle DAC.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Solution 18

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 19

Prove that a triangle ABC is isosceles, if:

(i)  altitude AD bisects angles BAC, or

(ii)  bisector of angle BAC is perpendicular to base BC.

Solution 19

(i)  In ΔABC, let the altitude AD bisects BAC.

Then we have to prove that the ΔABC is isosceles.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

In triangles ADB and ADC,

BAD = CAD (AD is bisector of BAC)

AD = AD (common)

ADB = ADC (Each equal to 90°)

ΔADB ΔADC (by ASA congruence criterion)

AB = AC (cpct)

Hence, ΔABC is an isosceles.

(ii)  In Δ ABC, the bisector of BAC is perpendicular to the base BC. We have to prove that the ΔABC is isosceles.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 In triangles ADB and ADC,

BAD = CAD (AD is bisector of BAC)

AD = AD (common)

ADB = ADC (Each equal to 90°)

ΔADB ΔADC (by ASA congruence criterion)

AB = AC (cpct)

Hence, ΔABC is an isosceles.

Question 20

In the given figure; AB = BC and AD = EC.

Prove that: BD = BE.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Solution 20

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

In ΔABC,

AB = BC (given)

BCA = BAC (Angles opposite to equal sides are equal)

BCD = BAE ….(i)

Given, AD = EC

AD + DE = EC + DE (Adding DE on both sides)

AE = CD ….(ii)

Now, in triangles ABE and CBD,

AB = BC (given)

BAE = BCD [From (i)]

AE = CD [From (ii)]

ΔABE ΔCBD

BE = BD (cpct)

Chapter 10 - Isosceles Triangle Exercise Ex. 10(B)

Question 1

If the equal sides of an isosceles triangle are produced, prove that the exterior angles so formed are obtuse and equal.

Solution 1

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Const: AB is produced to D and AC is produced to E so that exterior angles Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle is formed.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Since angle B and angle C are acute they cannot be right angles or obtuse angles.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore, exterior angles formed are obtuse and equal.

Question 2

In the given figure, AB = AC. Prove that:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i) DP = DQ

(ii) AP = AQ

(iii) AD bisects angle A

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Const: Join AD.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii) We have already proved that Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore,BP = CQ[cpct]

Now,

AB = AC[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAB - BP = AC - CQ

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAP = AQ

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Hence, AD bisects angle A.

Question 3

In triangle ABC, AB = AC; BE Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAC and CF Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAB. Prove that:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(i) BE = CF

(ii) AF = AE

Solution 3

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

(ii)Since Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 4

In isosceles triangle ABC, AB = AC. The side BA is produced to D such that BA = AD. Prove that: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCD = 90o

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Const: Join CD.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Adding (i) and (ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 5

(i) In triangle ABC, AB = AC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle= 36°. If the internal bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Trianglemeets AB at point D, prove that AD = BC.

(ii) If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

 

 

 

 

Question 6

Prove that the bisectors of the base angles of an isosceles triangle are equal.

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 7

In the given figure, AB = AC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleECB = 90o

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Prove that:

(i) BD = CE

(ii) AD = AE

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleECB = 90o[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleECB …….(ii)

Subtracting (i) from (ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 8

ABC and DBC are two isosceles triangles on the same side of BC. Prove that:

(i) DA (or AD) produced bisects BC at right angle.

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBDA = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCDA.

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

DA is produced to meet BC in L.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Subtracting (i) from (ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

From (iii), (iv) and (v)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

From (vi) and (vii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 9

The bisectors of the equal angles B and C of an isosceles triangle ABC meet at O. Prove that AO bisects angle A.

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC, we have AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC [angles opposite to equal sides are equal]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABO and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACO,

AB = AC [Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleOBC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleOCB [From (i)]

OB = OC [From (ii)]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore, AO bisects Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBAC.

Question 10

Prove that the medians corresponding to equal sides of an isosceles triangle are equal.

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 11

Use the given figure to prove that, AB = AC.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

From (i), (ii) and (iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 12

In the given figure; AE bisects exterior angle CAD and AE is parallel to BC.

Prove that: AB = AC.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 12

Since AE || BC and DAB is the transversal

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Since AE || BC and AC is the transversal

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

But AE bisects Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAB = AC[Sides opposite to equal angles are equal]

Question 13

In an equilateral triangle ABC; points P, Q and R are taken on the sides AB, BC and CA respectively such that AP = BQ = CR. Prove that triangle PQR is equilateral.

Solution 13

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AB = BC = CA…….(i) [Given]

AP = BQ = CR…….(ii) [Given]

Subtracting (ii) from (i)

AB - AP = BC - BQ = CA - CR

BP = CQ = AR …………(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle ……..(iv) [angles opp. to equal sides are equal]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

From (v) and (vi)

PQ = QR = PR

Therefore, PQR is an equilateral triangle.

Question 14

In triangle ABC, altitudes BE and CF are equal. Prove that the triangle is isosceles.

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABE and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACF,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA[Common]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAEB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAFC = 900[Given: BE Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAC; CF Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAB]

BE = CF[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore, ABC is an isosceles triangle.

Question 15

Through any point in the bisector of angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

Solution 15

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AL is bisector of angle A. Let D is any point on AL. From D, a straight line DE is drawn parallel to AC.

DE || AC [Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADE = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDAC….(i) [Alternate angles]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDAC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDAE…….(ii) [AL is bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA]

From (i) and (ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADE = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleDAE

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAE = ED [Sides opposite to equal angles are equal]

Therefore, AED is an isosceles triangle.

Question 16

In triangle ABC; AB = AC. P, Q and R are mid-points of sides AB, AC and BC respectively. Prove that:

(i) PR = QR(ii) BQ = CP

Solution 16

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC,

AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAP = AQ …….(i)[ Since P and Q are mid - points]

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCA,

PR = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[PR is line joining the mid - points of AB and BC]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePR = AQ……..(ii)

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAB,

QR = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[QR is line joining the mid - points of AC and BC]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleQR = AP……(iii)

From (i), (ii) and (iii)

PR = QR

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC

Also,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBPC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCQB,

BP = CQ

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC

BC = BC

Therefore, ΔBPCSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleΔCQB   [SAS]

BP = CP

Question 17

From the following figure, prove that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE

(ii) AD = CE

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Solution 17

(i) In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB,

AC = AC[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB …….(i)[angles opposite to equal sides are equal]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB = 1800 …….(ii)[DCB is a straight line]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE = 1800 ……..(iii)[ABE is a straight line]

Equating (ii) and (iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACB + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE[From (i)]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 18

Equal sides AB and AC of an isosceles triangle ABC are produced. The bisectors of the exterior angle so formed meet at D. Prove that AD bisects angle A.

Solution 18

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AB is produced to E and AC is produced to F. BD is bisector of angle CBE and CD is bisector of angle BCF. BD and CD meet at D.

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC,

AB = AC[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB[angles opposite to equal sides are equal]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE = 1800 - Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleB[ABE is a straight line]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[BD is bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCBE]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Similarly,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCF = 1800 - Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC[ACF is a straight line]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[CD is bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCF]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Now,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCD,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBD = CD

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABD and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD,

AB = AC[Given]

AD = AD[Common]

BD = CD[Proved]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore, AD bisectsSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA.

Question 19

ABC is a triangle. The bisector of the angle BCA meets AB in X. A point Y lies on CX such that AX = AY.

Prove that Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC.

Solution 19

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC,

CX is the angle bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCX ....... (i)

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAXY,

AX = AY [Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAXY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYX …….(ii) [angles opposite to equal sides are equal]

Now Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleXYC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAXB = 180° [straight line]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYX + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAXY + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBXY

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBXY ........ (iii) [From (ii)]

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBXC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAYC + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACY + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBXC + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBCX + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleXBC = 180°

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleXBC [From (i) and (iii)]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAY = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC

Question 20

In the following figure; IA and IB are bisectors of angles CAB and CBA respectively. CP is parallel to IA and CQ is parallel to IB.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Prove that:

PQ = The perimeter of the Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC.

Solution 20

Since IA || CP and CA is a transversal

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAI = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePCA [Alternate angles]

Also, IA || CP and AP is a transversal

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleIAB = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAPC [Corresponding angles]

But Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleCAI = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleIAB [Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TrianglePCA = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAPC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAC = AP

Similarly,

BC = BQ

Now,

PQ = AP + AB + BQ

= AC + AB + BC

= Perimeter of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC

Question 21

Sides AB and AC of a triangle ABC are equal. BC is produced through C upto a point D such that AC = CD. D and A are joined and produced upto point E. If angle BAE = 108o; find angle ADB.

Solution 21

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABD,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBAE = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle3 + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle1080 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle3 + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

But AB = AC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle3 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle2

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle1080 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle2 + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB ……(i)

Now,

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACD,

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle2=Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle1+ Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

But AC = CD

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle1 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle2 = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle2 = 2Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

Putting this value in (i)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle1080 = 2Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB + Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle3Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB = 1080

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADB = 360

Question 22

The given figure shows an equilateral triangle ABC with each side 15 cm. Also, DE//BC, DF//AC and EG//AB.

 

 

If DE + DF + EG = 20 cm, find FG.

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 

Solution 22

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 23

If all the three altitudes of a triangle are equal, the triangle is equilateral. Prove it.

Solution 23

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

In right Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBEC and Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBFC,

BE = CF[Given]

BC = BC[Common]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBEC = Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleBFC[each = 900]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Therefore, ABC is an equilateral triangle.

Question 24

In a Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC, the internal bisector of angle A meets opposite side BC at point D. Through vertex C, line CE is drawn parallel to DA which meets BA produced at point E. Show that Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACE is isosceles.

Solution 24

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

DA || CE[Given]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[Corresponding angles]

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[Alternate angles]

But Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle[ AD is the bisector of Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleA]

From (i), (ii) and (iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleAC = AE

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleACE is an isosceles triangle.

Question 25

In triangle ABC, bisector of angle BAC meets opposite side BC at point D. If BD = CD, prove that Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC is isosceles.

Solution 25

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Produce AD upto E such that AD = DE.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Hence, ABC is an isosceles triangle.

Question 26

In Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleABC, D is point on BC such that AB = AD = BD = DC. Show that:

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleADC: Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles TriangleC = 4: 1.

Solution 26

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Since AB = AD = BD

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangleis an equilateral triangle.

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Again inSelina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

AD = DC

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

Question 27

Using the information given in each of the following figures, find the values of a and b. 

 

 

 Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 

 

 [Given: CE = AC] 

 

 

 

 

 

 Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 

Solution 27

(i)

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle 

 

 

(ii)

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Isosceles Triangle