Asked in: March 2024 and June 2026 board papersAB 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 \…

Answer: (b) \(\angle CAT = 90^\circ\), \(\angle ABC = 90^\circ\) (c) Yes, they are congruent.

(a) [Figure: circle with centre O; diameter AOC; chord AB with B on the circle; line AT touches the circle at A; seg BC joined.]

(b) Seg OA is a radius and AT is the tangent at A, so \(\angle OAT = 90^\circ\) (tangent theorem). O lies on AC, so \(\angle CAT = 90^\circ\).

\(\angle ABC\) is inscribed in the semicircle ABC (AC is a diameter), so \(\angle ABC = 90^\circ\) (angle inscribed in a semicircle).

(c) Both angles measure 90°, so \(\angle CAT \cong \angle ABC\). This is also an example of the tangent secant angle theorem: \(\angle CAT = \dfrac{1}{2} m(\text{arc ABC}) = \dfrac{1}{2} \times 180^\circ = 90^\circ\).