Question

\( AB \) is a chord of a circle with center \( O \), \( AOC \) is the diameter of the circle, and \( AT \) is a tangent at \( A \). Write the answers to the following questions:
[(a)] Draw the figure using the given information.
[(b)] Find the measures of \( \angle CAT \) and \( \angle ABC \) with reasons.
[(c)] Determine whether \( \angle CAT \) and \( \angle ABC \) are congruent. Justify your answer.

Hint:

For tangents and diameters in circles, remember that the tangent at a point of contact is perpendicular to the radius, and the angle in a semicircle is always \( 90^\circ \).

Answer 1

(a) Drawing the figure:
The figure is shown above, where \( AOC \) is the diameter, \( AB \) is a chord, and \( AT \) is a tangent.
(b) Measures of \( \angle CAT \) and \( \angle ABC \):
\[ \angle CAT = 90^\circ \quad \text{(Tangent-radius property: tangent is perpendicular to the radius at the point of contact).} \] \[ \angle ABC = 90^\circ \quad \text{(Angle subtended by the diameter in a semicircle is a right angle).} \] (c) Congruence of \( \angle CAT \) and \( \angle ABC \):
Yes, \( \angle CAT \) and \( \angle ABC \) are congruent as they are both equal to \( 90^\circ \) by different properties (tangent-radius property and semicircle theorem).