Question

Noah is baking a two-layer cake, in which the bottom layer is a circle and the top layer is a triangle. If segment AB = 6 inches and arc AB≅ arc AC, what does Noah know about the top layer of his cake?
Circumscribed circle D around triangle ABC, arcs AB and AC are congruent.
A. segment AB is twice the length of segment BC because their arcs are congruent; therefore, ΔABC is an equilateral triangle.
B. segment AB is twice the length of segment BC because their arcs are congruent; therefore, ΔABC is an isosceles triangle.
C. segment AB ≅ segment BC because their arcs are congruent; therefore, ΔABC is an equilateral triangle.
D. segment AB ≅segment BC because their arcs are congruent; therefore, ΔABC is an isosceles triangle.

Answer 1

Answer:

segment AB ≅ segment BC because their arcs are congruent; therefore, ΔABC is an isosceles triangle ⇒ answer D

Step-by-step 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.