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.

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

isosceles triangle.

The statement that gives what Noah knows is option D.

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 = 6 inches

≅

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 , which is segment is equal to the chord that subtends arc which is chord

= , therefore, ≅ , and ΔABC is an isosceles triangle

What is known from the given information is therefore;

Learn more about isosceles triangles here:

link

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.