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: 