Question

Noah is baking a two-layer cake, in which the bottom layer is a circle and the top layer is a triangle. AB = 10 inches and arcAB ≅ arcAC, what does Noah know about the top layer of his cake?
Answers-
A. AB is twice the length of AC because their arcs are congruent, therefore, angle ABC is an equilateral triangle.
B. AB ≅ AC because their arcs are congruent, therefore, angle ABC is an isosceles triangle.
C. AB is twice the length of AC because their arcs are congruent, therefore angle ABC is an isosceles triangle.
D. AB ≅ AC because their arcs are congruent, therefore, angle ABC is an equilateral triangle.

Answer 1

Answer

Hello There. The correct answer is D. AB ≅ AC because their arcs are congruent, therefore, angle ABC is an equilateral triangle.

Explanation

* Lets revise some facts in the circle

- If a circle is circumscribed around a triangle, then the vertices of the

 triangle divide the circle into three arcs

- The three sides of the triangle are the chords of the circle

- Each chord in the circle subtended by an arc

- If two arcs equal, then they are subtended equal chords

* Lets solve the problem

∵ Circumscribed circle D around triangle ABC

- The vertices of triangle ABC lie on the circumference of the circle

∴ AB , BC , AC are the chords of the circle

- Arcs AB and AC are congruent

∵ Arc AB ≅ arc AC

∵ Arc AB subtended the chord AB

∵ Arc AC subtended the chord AC

∴ AB = AC

∵ AB = 6 inches

∴ AC = 6 inches

- The triangle with two equal sides is called an isosceles triangle

∴ Δ ABC is an isosceles triangle

∴ Noah know that the top layer of his cake is an isosceles triangle

* segment AB ≅ segment BC because their arcs are congruent;

 therefore,  ΔABC is an isosceles triangle.

Hope it helps!

ItsNobody