SELINA Solutions for Class 9 Maths Chapter 24 - Solution of Right Triangles [Simple 2-D Problems Involving One Right-angled Triangle]

Get free access to Selina Solutions for ICSE Class 9 Mathematics Chapter 24 Solution of Right Triangles, Simple 2D problems involving one right-angled Triangle on TopperLearning. Learn how to calculate the lengths of diagonals of a rhombus. Also, learn how to find the length of side of figure formed using different triangles and a quadrilateral.

In this ICSE Class 9 Maths chapter, understand the ways to use right-angled triangle to find answers involving a complex figure. If you have any Math doubts, you can anytime post your question on our ‘UnDoubt’ platform for explanations and it will be answered by Math experts.

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Chapter 24 - Solution of Right Triangles [Simple 2-D Problems Involving One Right-angled Triangle] Exercise Ex. 24

Question 1

Fin 'x' if:

(i)

(ii)

(iii)

Solution 1

(i)

From the figure we have

(ii)

From the figure we have

(iii)

From the figure we have

Question 2

Find angle 'A' if:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 2

(i)

From the figure we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

From the figure we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(iii)

From the figure we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 3

Find angle 'x' if:

Solution 3

The figure is drawn as follows:

The above figure we have

Again

Now

Question 4

Find AD, if:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 4

(i)

From the right triangle ABE

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore AE = BE = 50 m.

Now from the rectangle BCDE we have

DE = BC = 10 m.

Therefore the length of AD will be:

AD = AE + DE = 50 + 10 = 60 m.

(ii)

From the triangle ABD we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 5

Find the length of AD.

Given: Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleABC = 60o.

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleDBC = 45o

And BC = 40 cm.

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

 

 

Question 6

Find the lengths of diagonals AC and BD. Given AB = 60 cm and BAD = 60o.

Solution 6

We know, diagonals of a rhombus bisect each other at right angles and also bisect the angle of vertex.

The figure is shown below:

Now

And

Also given

In right triangle AOB

Also

Therefore,

.

.

Question 7

Find AB.

Solution 7

Consider the figure

From right triangle ACF

From triangle DEB

Given , So

Therefore

Thus AB = AC + CD + BD = 54.64 cm.

Question 8

In trapezium ABCD, as shown, AB // DC, AD = DC = BC = 20 cm and A = 60o. Find:

(i) length of AB

(ii) distance between AB and DC.

Solution 8

First draw two perpendiculars to AB from the point D and C respectively. Since AB|| CD therefore PMCD will be a rectangle.

Consider the figure,

(i)

From right triangle ADP we have

Similarly from the right triangle BMC we have BM = 10 cm.

Now from the rectangle PMCD we have CD = PM = 20 cm.

Therefore

AB = AP + PM + MB = 10 + 20 + 10 = 40 cm.

(ii)

Again from the right triangle APD we have

Therefore the distance between AB and CD is .

Question 9

Use the information given to find the length of AB.

Solution 9

From right triangle AQP

Also from triangle PBR

Therefore,

AB = AP + PB =.

Question 10

Find the length of AB.


Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 11

In the given figure, AB and EC are parallel to each other. Sides AD and BC are 2 cm each and are perpendicular to AB.

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Given that Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleAED = 60o and Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleACD = 45o. Calculate:

(i) AB(ii) AC(iii) AE

Solution 11

(i)

From the triangle ADC we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Since AD || DC andSelina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle,  ABCD is a parallelogram and hence opposite sides are equal.

Therefore AB = DC = 2 cm

(ii)

Again

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(iii)

From the right triangle ADE we have

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 12

In the given figure, B = 60° , AB = 16 cm and BC = 23 cm,

Calculate:

(i) BE (ii) AC

 

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle 

 

 

Solution 12

In ∆ABE,

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(i) In ∆ABE, mAEB = 90° 

By Pythagoras Theorem, we get

BE2 = AB2 - AE2

BE2 = (16)2 - (Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle)2

BE2 = 256 - 192

BE2 = 64

BE = 8cm

(ii) EC = BC - BE = 23 - 8 = 15

In ∆AEC, mAEC = 90° 

By Pythagoras Theorem, we get

AC2 = AE2 + EC2

AC2 = (Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle)2 + (15)2

AC2 = 192 + 225

AC2 = 147

AC = 20. 42 cm

Question 13

Find

(i) BC

(ii) AD

(iii) AC

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 13

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle


Question 14

In right-angled triangle ABC; B = 90o. Find the magnitude of angle A, if:

(i) AB is Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangletimes of BC.

(ii) BC is Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangletimes of AB.

Solution 14

Consider the figure

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(i) Here AB is Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangletimes of BC means

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

Again from the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore, magnitude of angle A is 30Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 15

A ladder is placed against a vertical tower. If the ladder makes an angle of 30o with the ground and reaches upto a height of 15 m of the tower; find length of the ladder.

Solution 15

Given that the ladder makes an angle of 30o with the ground and reaches upto a height of 15 m of the tower which is shown in the figure below:

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Suppose the length of the ladder is x m

From the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore the length of the ladder is 30m.

Question 16

A kite is attached to a 100 m long string. Find the greatest height reached by the kite when its string makes an angles of 60o with the level ground.

Solution 16

Given that the kite is attached to a 100 m long string and it makes an angle of  60Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle with the ground level which is shown in the figure below:

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Suppose that the greatest height is x m.

From the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore the greatest height reached by the kite is Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle.

Question 17

Find AB and BC, if:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 17

(i)Let  BC = xm

BD = BC + CD = (x+20)cm

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleABD,

tan 30Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

x+20 = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleAB        .....(1)

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleABC

tan 45Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

1 = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

AB = x       ... (2)

 

From (1)

AB + 20 = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleAB

AB(Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle-1) = 20

ABSelina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

    = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

    = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle = 27.32 cm

From (2)

AB = x = 27.32cm

Therefore BC = x = AB = 27.32cm

Therefore, AB = 27.32cm, BC = 27.32cm

(ii)

Let  BC = xm

BD = BC + CD = (x + 20) cm

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle,

tan 30Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

x + 20 = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle AB       ...(1)

 

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

tan 60Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

x = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle                   ...(2 )

 

From (1)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle


From (2)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore BC = x = 10cm

 

Therefore,

AB = 17.32cm, BC = 10cm

 

(iii)

Let  BC = xm

BD = BC + CD = (x + 20)cm

 

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle,

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

x + 20 = AB         ...(1)

 

In Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

 

From (1)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

From (2)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore,

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 18

Find PQ, if AB = 150 m, Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleP = 30o and Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleQ = 45o.

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 18

(i) From Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Also, from Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleABQ

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore,

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

(ii) From Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Also, from Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleABQ

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore,

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Question 19

If tan xo =Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle,

tan yo = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangleand AB = 48 m; find the length of CD.

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Solution 19

Given tan xo =Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangletan to = Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangleand AB = 48 m;

Let length of  BC = xm

From Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleADC

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Also, from Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled TriangleBDC

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

From (1)

Selina Solutions Icse Class 9 Mathematics Chapter - Solution Of Right Triangles Simple 2 D Problems Involving One Right Angled Triangle

Therefore, length of CD is 45 m.

Question 20

The perimeter of a rhombus is 96 cm and obtuse angle of it is 120o. Find the lengths of its diagonals.

Solution 20

Since in a rhombus all sides are equal.

The diagram is shown below:

Therefore ,Let .

We also know that in rhombus diagonals bisect each other perpendicularly and diagonal bisect the angle at vertex.

Hence POR is a right angle triangle and

But

Therefore,

Also,

But

Therefore,

So, the length of the diagonal and