AP State Syllabus AP Board 9th Class Maths Solutions Chapter 12 Circles Ex 12.5 Textbook Questions and Answers.

## AP State Syllabus 9th Class Maths Solutions 12th Lesson Circles Exercise 12.5

Question 1.

Find the values of x and y in the figures given below.

i)

Solution:

From the figure x = y [ ∵ angles opp. to opp. to equal sides]

But x + y + 30° = 180°

∴ x + y = 180° – 30° = 150°

⇒ x + y = \(\frac { 150° }{ 2 }\) = 75°

ii)

Solution:

From the figure x° + 110° = 180°

[ ∵ Opp. angles of a cyclic quad, are supplementary]

y + 85°= 180°

∴ x= 180° – 110°; y = 180° – 85°

x = 70°; y = 95°

iii)

Solution:

From the figure x = 90° [angle in a semi-circle]

∴ y = 90° – 50° [∵ angle sum property]

= 40°

Question 2.

Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ∠A + ∠C = 180°, then prove that the vertex D also lie on the same circle.

Solution:

Given : ∠A + ∠C = 180°

∴ ∠B + ∠D = 360° – 180°

[ ∵ sum of the four angles of a quad. is 360 ].

Now in □ABCD, sum of the pairs of opp. angles is 180°.

∴ □ABCD must be a cyclic quadrilateral, i.e., D also lie on the same circle on which the vertices A, B and C lie. Hence proved.

Question 3.

Prove that a cyclic rhombus is a square.

Solution:

Let □ABCD be a cyclic rhombus,

i.e., AB = BC = CD = DA and

∠A + ∠C = ∠B + ∠D = 180°

But a rhombus is basically a parallelo-gram.

Question 4.

For each of the following, draw a circle and inscribe the figure given. If a poly¬gon of the given type can’t be in-scribed, write not possible.

a) Rectangle

b) Trapezium .

c) Obtuse triangle

d) Non-rectangular parallelogram

e) Acute isosceles triangle

f) A quadrilateral PQRS with PR as diameter.

Solution: