Question

1. triangle abc is inscribed in circle o as shown. (overline{ac}) is a diameter of circle o. which relationship must be true? (a) (ao = ab) (b) (mangle b=90^{circ}) (c) (ab + bc = ac) (d) (mangle a + mangle b + mangle c=360^{circ})

Explanation:

Step1: Recall circle - triangle property

If a triangle is inscribed in a circle and one of its sides is the diameter of the circle, then the angle opposite the diameter is a right - angle.

Step2: Analyze each option

• Option A: $AO$ is the radius of the circle and $AB$ is a chord. There is no reason for $AO = AB$ in general.
• Option B: Since $\triangle ABC$ is inscribed in circle $O$ with $AC$ as the diameter, by the Thales' theorem, $\angle B=90^{\circ}$, so $m\angle B = 90^{\circ}$.
• Option C: By the triangle inequality, in $\triangle ABC$, $AB + BC>AC$.
• Option D: The sum of the interior angles of a triangle is $180^{\circ}$, so $m\angle A + m\angle B+m\angle C = 180^{\circ}$, not $360^{\circ}$.

Answer:

B. $m\angle B = 90^{\circ}$