Question

question 8(multiple choice worth 1 points)
(07 12 mc)
noah is baking a two-layer cake, in which the bottom layer is a circle and the top layer is a triangle. if ab = 10 inches and ab ≅ ac, what does noah know about the top layer of his cake?
ab ≅ ac because their arcs are congruent, therefore, △abc is an isosceles triangle
ab ≅ ac because their arcs are congruent, therefore, △abc is an equilateral triangle
ab is twice the length of ac because their arcs are congruent, therefore, △abc is an equilateral triangle
ab is twice the length of ac because their arcs are congruent, therefore, △abc is an equilateral triangle

Explanation:

Step1: Recall the property of congruent arcs

In a circle, if two arcs are congruent (\(\overset{\frown}{AB}\cong\overset{\frown}{AC}\)), then the chords subtended by these arcs are congruent. The formula for the relationship between an arc and its subtended chord is that congruent arcs in the same circle (or congruent circles) have congruent chords. So, if \(\overset{\frown}{AB}\cong\overset{\frown}{AC}\), then \(AB = AC\).

Step2: Determine the type of triangle

A triangle with two congruent sides (\(AB = AC\)) is an isosceles triangle. The definition of an isosceles triangle is a triangle that has at least two congruent sides.

Answer:

\(\overset{\frown}{AB}\cong\overset{\frown}{AC}\) implies \(AB = AC\) (chords of congruent arcs are congruent), so \(\triangle ABC\) is an isosceles triangle. The correct option is “\(\overline{AB}=\overline{AC}\) because their arcs are congruent, therefore, \(\triangle ABC\) is an isosceles triangle”.