AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.Write answers of the following questions :(a) Draw the figure using given information.(b) Find the measures of ∠CAT and ∠ABC with reasons.(c) Whether ∠CAT and ∠A…

Answer: (b) m∠CAT = 90º and m∠ABC = 90º. (c) Yes, ∠CAT ≅ ∠ABC, since both measure 90º.

(a) Draw a circle with centre O and a diameter AOC. Take a point B on the circle and join AB, BC to get chord AB. Draw the tangent AT at A.

(b) AT is a tangent at A and OA is the radius through the point of contact, so OA ⊥ AT. As A, O, C are collinear, AC lies along OA; hence ∠CAT = 90º.

∠ABC is an angle inscribed in a semicircle (AC is a diameter), so ∠ABC = 90º.

(c) Yes. ∠CAT = 90º and ∠ABC = 90º, so ∠CAT ≅ ∠ABC. (Each is a right angle — one is the angle between a tangent and the radius at the point of contact, the other is an angle in a semicircle.)