SELINA Solutions for Class 9 Maths Chapter 13 - Pythagoras Theorem [Proof and Simple Applications with Converse]

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Chapter 13 - Pythagoras Theorem [Proof and Simple Applications with Converse] Exercise Ex. 13(A)

Question 1

A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.

Solution 1

The pictorial representation of the given problem is given below,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

(i)Here, AB is the hypotenuse. Therefore applying the Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Therefore, the distance of the other end of the ladder from the ground is 12m

Question 2

A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.

Solution 2

Here , we need to measure the distance AB as shown in the figure below,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse






Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Therefore , in this case

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Therefore the required distance is 64.03 m.

Question 3

In the figure: Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConversePSQ = 90o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 3

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Now, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

The length of PRSelina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 4

The given figure shows a quadrilateral ABCD in which AD = 13 cm, DC = 12 cm, BC = 3 cm and Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseABD = Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseBCD = 90o. Calculate the length of AB.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 4

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Now, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

The length of AB is 4 cm.

Question 5

AD is drawn perpendicular to base BC of an equilateral triangle ABC. Given BC = 10 cm, find the length of AD, correct to 1 place of decimal.

Solution 5

Since ABC is an equilateral triangle therefore, all the sides of the triangle are of same measure and the perpendicular AD will divide BC in two equal parts.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Here, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseTherefore, the length of AD is 8.7 cm

Question 6

In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC= 3 cm. Calculate the length of OC.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 6

We have Pythagoras theorem which states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

Now, from (i) and(ii),

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 7

In triangle ABC,

AB = AC = x, BC = 10 cm and the area of the triangle is 60 cm2. Find x.

Solution 7

Here, the diagram will be,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse










We have Pythagoras theorem which states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Since, ABC is an isosceles triangle, therefore perpendicular from vertex will cut the base in two equal segments.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Therefore, x is 13cm

Question 8

If the sides of triangle are in the ratio 1 : Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse: 1, show that is a right-angled triangle.

Solution 8

Let, the sides of the triangle be, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Now, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Here, in (i) it is shown that, square of one side of the given triangle is equal to the addition of square of other two sides. This is nothing but Pythagoras theorem which states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Therefore, the given triangle is a right angled triangle.

Question 9

Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m; find the distance between their tips.

Solution 9

The diagram of the given problem is given below,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

We have Pythagoras theorem which states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 10

In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm; find the length of side BC.

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

Solution 10

 

 

 

Take M be the point on CD such that AB = DM.

 

 

So DM = 7cm and MC = 10 cm

 

 

Join points B and M to form the line segment BM.

 

 

So BM || AD also BM = AD.

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

Question 11

In the given figure, B = 90°, XY || BC, AB = 12cm, AY = 8cm and AX: XB = 1: 2 = AY: YC. Find the lengths of AC and BC.


Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 11

Given that AX: XB = 1: 2 = AY: YC.

Let x be the common multiple for which this proportion gets satisfied.

So, AX = 1x and XB = 2x

AX + XB = 1x + 2x = 3x

AB = 3x .….(A - X - B)

12 = 3x

x = 4

AX = 1x = 4 and XB = 2x = 2 × 4 = 8

Similarly,

AY = 1y and YC = 2y

AY = 8…(given)

8 = y

YC = 2y = 2 × 8 = 16

AC = AY + YC = 8 + 16 = 24 cm

∆ABC is a right angled triangle. …. Given

By Pythagoras Theorem, we get

AB2 + BC2 = AC2

BC2 = AC2 - AB2

BC2 = (24)2 - (12)2

BC2 = 576 - 144

BC2 = 432

BC= Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Conversecm

AC = 24 cm and BC =Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Conversecm

Question 12

In ΔABC, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse Find the sides of the triangle, if:

 

 

(i) AB = (x - 3) cm, BC = (x + 4) cm and AC = (x + 6) cm

 

 

(ii) AB = x cm, BC = (4x + 4) cm and AC = (4x + 5) cm

 

Solution 12

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

Chapter 13 - Pythagoras Theorem [Proof and Simple Applications with Converse] Exercise Ex. 13(B)

Question 1

In the figure, given below, AD Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse BC. Prove that: c2 = a2 + b2 - 2ax.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 1

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse and applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse and applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

From (i) and (ii) we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Hence Proved.

Question 2

In equilateral Δ ABC, AD Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse BC and BC = x cm. Find, in terms of x, the length of AD.

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

In equilateral Δ ABC, AD Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse BC.

Therefore, BD = DC = x/2 cm.

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

Question 3

ABC is a triangle, right-angled at B. M is a point on BC. Prove that:

AM2 + BC2 = AC2 + BM2.

Solution 3

The pictorial form of the given problem is as follows,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse








Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse





Now, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse




From (i) and (ii) we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 4

M andN are the mid-points of the sides QR and PQ respectively of a Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConversePQR, right-angled at Q. Prove that:

(i) PM2 + RN2 = 5 MN2

(ii) 4 PM2 = 4 PQ2 + QR2

(iii) 4 RN2 = PQ2 + 4 QR2

(iv) 4 (PM2 + RN2) = 5 PR2

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse












We draw , PM,MN,NR

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Since, M andN are the mid-points of the sides QR and PQ respectively, therefore, PN=NQ,QM=RM

(i)

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse












Now, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras

theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Adding (i) and (ii) we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

(ii)

We consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse.

(iii)

We consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse










(iv)

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse












Now, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Adding (i) and (ii) we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 5

In triangle ABC, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseB = 90o and D is the mid-point of BC. Prove that: AC2 = AD2 + 3CD2.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

In triangle ABC, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseB = 90o and D is the mid-point of BC. Join AD. Therefore, BD=DC

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Similarly, we get from rt. angle triangles ABC we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

From (i) and (ii) ,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 6

In a rectangle ABCD, prove that:

AC2 + BD2 = AB2 + BC2 + CD2 + DA2.

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Since, ABCD is a rectangle angles A,B,C and D are rt. angles.

First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Similarly, we get from rt. angle triangle BDC we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Adding (i) and (ii) ,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 7

In a quadrilateral ABCD, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseB = 900 and Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With ConverseD = 900. Prove that: 2AC2 - AB2 = BC2 + CD2 + DA2

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

Question 8

O is any point inside a rectangle ABCD. Prove that: OB2 + OD2 = OC2 + OA2.

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

Draw rectangle ABCD with arbitrary point O within it, and then draw lines OA, OB, OC, OD. Then draw lines from point O perpendicular to the sides: OE, OF, OG, OH.

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

Using Pythagorean theorem we have from the above diagram:

OA2 = AH2 + OH2 = AH2 + AE2

OC2 = CG2 + OG2 = EB2 + HD2

OB2 = EO2 + BE2 = AH2 + BE2

OD2 = HD2 + OH2 = HD2 + AE2

Adding these equalities we get:

OA2 + OC2 = AH2 + HD2 + AE2 + EB2

OB2 + OD2 = AH2 + HD2 + AE2 + EB2

From which we prove that for any point within the rectangle there is the relation

OA2 + OC2 = OB2 + OD2

Hence Proved.

Question 9

In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC. Prove that:

AR2 + BP2 + CQ2 = AQ2 + CP2 + BR2

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 9

Here, we first need to join OA, OB, and OC after which the figure becomes as follows,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides. First, we consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Similarly, from triangles, BPO,COQ,AOQ,CPO and BRO we get the following results,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Adding (i), (ii) and (iii),we get 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Adding (iv), (v) and (vi),we get ,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

From (vii) and (viii), we get,

AR2 + BP2 + CQ2 = AQ2 + CP2 + BR2

Hence proved.

Question 10

Diagonals of rhombus ABCD intersect each other at point O. Prove that:

OA2 + OC2 = 2AD2 - Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Diagonals of the rhombus are perpendicular to each other.

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

 

Question 11

In the figure AB = BC and AD is perpendicular to CD. Prove that:

AC2 = 2BC. DC.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 11

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

We consider the Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 12

In an isosceles triangle ABC; AB = AC and D is point on BC produced. Prove that:

AD2 = AC2 + BD.CD.

Solution 12

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

In an isosceles triangle ABC; AB = AC and D is point on BC produced. Construct AE perpendicular BC.

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

We consider the rt. angled Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Question 13

In triangle ABC, angle A = 90o, CA = AB and D is point on AB produced. Prove that DC2 -BD2 = 2AB.AD.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Solution 13

Pythagoras theorem states that in a right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.

We consider the rt. angled Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converseand applying Pythagoras theorem we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Similarly, in Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Putting, Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Conversefrom (ii) in (i) we get,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Hence Proved.

Question 14

In triangle ABC, AB = AC and BD is perpendicular to AC. Prove that: BD2 - CD2 = 2CD × AD.

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

 

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse 

Question 15

In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse

Prove that : 2AC2 = 2AB2 + BC2

Solution 15

Here,

Selina Solutions Icse Class 9 Mathematics Chapter - Pythagoras Theorem Proof And Simple Applications With Converse