# 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.

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

Fin 'x' if:

(i)

(ii)

(iii)

(i)

From the figure we have

_{}

(ii)

From the figure we have

_{}

(iii)

From the figure we have

_{}

Find angle 'A' if:

(i)

(ii)

(iii)

(i)

From the figure we have

_{}

(ii)

From the figure we have

_{}

(iii)

From the figure we have

_{}

Find angle 'x' if:

The figure is drawn as follows:

The above figure we have

_{}

Again

_{}

Now

_{}

Find AD, if:

(i)

(ii)

(i)

From the right triangle ABE

_{}

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

Find the length of AD.

Given: _{}ABC = 60^{o}.

_{}DBC = 45^{o}

And BC = 40 cm.

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

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,

_{}.

_{}.

Find AB.

Consider the figure

From right triangle ACF

_{}

From triangle DEB

_{}

Given
_{}, So _{}

Therefore

_{}

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

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

(i) length of AB

(ii) distance between AB and DC.

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 _{}.

Use the information given to find the length of AB.

From right triangle AQP

_{}

Also from triangle PBR

_{}

Therefore,

AB
= AP + PB =_{}.

Find the length of AB.

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.

Given that _{}AED = 60^{o} and _{}ACD = 45^{o}. Calculate:

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

(i)

From the triangle ADC we have

_{}

Since AD || DC and_{}, ABCD is a parallelogram and hence opposite sides are equal.

Therefore AB = DC = 2 cm

(ii)

Again

_{}

(iii)

From the right triangle ADE we have

_{}

In
the given figure, ∠B = 60^{°} , AB = 16 cm and BC = 23 cm,

Calculate:

(i) BE (ii) AC

In ∆ABE,

(i) In ∆ABE, m∠AEB = 90^{°}

∴ By Pythagoras Theorem, we get

BE^{2} =
AB^{2} - AE^{2}

⇒ BE^{2}
= (16)^{2} - ()^{2}

⇒ BE^{2}
= 256 - 192

⇒ BE^{2}
= 64

⇒ BE = 8cm

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

In ∆AEC, m∠AEC
= 90^{°}

∴ By Pythagoras Theorem, we get

AC^{2} =
AE^{2} + EC^{2}

⇒ AC^{2}
= ()^{2} + (15)^{2}

⇒ AC^{2}
= 192 + 225

⇒ AC^{2}
= 147

⇒ AC = 20. 42 cm

Find

(i) BC

(ii) AD

(iii) AC

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

(i) AB is _{}times of BC.

(ii) BC is _{}times of AB.

Consider the figure

(i) Here AB is _{}times of BC means

_{}

(ii)

Again from the figure

_{}

Therefore, magnitude of angle A is 30

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

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

Suppose the length of the ladder is *x *m

From the figure

_{}

Therefore the length of the ladder is 30m.

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 60^{o} with the level ground.

Given that the kite is attached to a 100 m long string and it makes an angle of 60 with the ground level which is shown in the figure below:

Suppose that the greatest height is *x* m.

From the figure

_{}

Therefore the greatest height reached by the kite is _{}.

Find AB and BC, if:

(i)

(ii)

(iii)

(i)Let *BC = xm*

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

In *ABD*,

tan 30 =

x+20 = *AB * .....(1)

In *ABC*

tan 45 =

1 =

*AB = x * ... (2)

From (1)

*AB* + 20 = AB

*AB*(-1) = 20

*AB* =

=

= = 27.32* cm*

From (2)

_{AB = x = 27.32cm}

_{Therefore BC = x = AB = 27.32cm}

Therefore,* AB* = 27.32*cm*, *BC* = 27.32*cm*

(ii)

Let *BC = xm*

_{BD = BC + CD = (x + 20) cm}

In ,

tan 30 =

x + 20 = *AB * ...(1)

In

tan 60 =

x = ...(2 )

From (1)

From (2)

Therefore *BC* = *x* = 10*cm*

Therefore,

*AB* = 17.32*cm*, *BC* = 10*cm*

(iii)

Let * BC = xm*

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

In ,

x + 20 = *AB *...(1)

In

From (1)

_{}

From (2)

_{}

_{}

Therefore,

_{}

Find PQ, if AB = 150 m, _{}P = 30^{o} and _{}Q = 45^{o}.

(i)

(ii)

(i) From _{}

_{}

Also, from *ABQ*

_{}

Therefore,

_{}

(ii) From _{}

_{}

Also, from *ABQ*

_{}

Therefore,

_{}

If tan x^{o} =_{},

tan y^{o} = _{}and AB = 48 m; find the length of CD.

Given tan x^{o} =_{}tan t^{o} = _{}and AB = 48 m;

Let length of *BC = xm*

From *ADC*

_{}

Also, from *BDC*

_{}

From (1)

_{}

Therefore, length of CD is 45 *m*.

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

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 _{}

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