Q2 of 63 Page 150

In figure, ∠ABC = 45°. Then prove that OA ⊥ OC.

We know that angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle.

⇒∠AOC = 2∠ABC

Given ∠ABC = 45°

⇒∠AOC = 2(45°)

⇒∠AOC = 90°

Since ∠AOC = 90°, OA ⊥ OC.

Hence proved.

Prove that the line segment joining the mid-points of two parallel chords of a circle passes through the centre of the circle.

Write True/False for each and write the reason also for your answer.

(i) The angles subtended at any two points lying of the circle by every chord are equal.

(ii) In figure, AB is a diameter of a circle and C is any point on the circle. Then AC² + BC² = AB².

(iii) In figure 12.48, if ∠ADE = 120°, then ∠EAB = 30°.

(iv) In figure 12.48, ∠CAD = ∠CED.

O is the circumcircle of triangle ABC and D is the mid-point of the base BC. Prove that ∠BOD = ∠A.

On a common hypotenuse AB two right angled triangles ACB and ADB are drawn such that they lie on the opposite sides. Prove that ∠BAC = ∠BDC.