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.

Answer 2

The legs of an isosceles triangle are equal, therefore, ΔABC is an

isosceles triangle.

The statement that gives what Noah knows is option D.

• D. Segment $\overline{AB}$ ≅ segment $\overline{BC}$ because their arcs are congruent; therefore, ΔABC is an isosceles triangle

Reasons why the above option is correct

The given information are;

Number of layers in the cake = 2

Shape of the top layer of the cake = Triangle

Shape of the bottom layer of the cake = Circle

Length of segment $\overline{AB}$ = 6 inches

$\widehat{AB}$ ≅ $\widehat{AC}$

The center of the circumscribed circle = Point D

Required:

What Noah knows about the top layer (the triangle ΔABC)

Solution:

According to equal chords in equal circle theorem, we have that if two

arcs are equal, the chords by which they are subtended are also equal.

Therefore;

The chord that subtend arc $\widehat{AB}$, which is segment $\overline{AB}$ is equal to the chord that subtends arc $\mathbf{\widehat{AC}}$ which is chord $\overline{AC}$

$\overline{AB}$ = $\overline{AC}$, therefore, $\overline{AB}$ ≅ $\overline{AC}$, and ΔABC is an isosceles triangle

What is known from the given information is therefore;

• Segment $\underline{\overline{AB} \cong \overline{AC}}$ because their arcs are congruent; therefore, ΔABC is an isosceles triangle.

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