### Video Transcript

If 𝐴𝐵𝐶 is an isosceles triangle
with 𝐴𝐵 equal to two centimeters and 𝐵𝐶 equal to five centimeters, find
𝐴𝐶.

To begin, we will recall the
definition of an isosceles triangle. An isosceles triangle is a triangle
that has two congruent sides. The congruent sides are called the
legs of the triangle, and the third side is called the base. The following triangle diagrams are
not drawn to scale, but they can help us think through the possible arrangement of
sides in this triangle. For triangle 𝐴𝐵𝐶 to be
isosceles, the missing side 𝐴𝐶 must be congruent to either side 𝐴𝐵 or side
𝐵𝐶. If 𝐴𝐵 is congruent to 𝐴𝐶, then
𝐴𝐶 equals two. If 𝐵𝐶 is congruent to 𝐴𝐶, then
𝐴𝐶 has to equal five.

To determine whether the third side
length is two centimeters or five centimeters, we will need to recall the triangle
inequality theorem. The triangle inequality tells us
that the sum of the lengths of any two sides of a triangle must be greater than the
length of the third side. And if this holds true, then the
triangle can be constructed. Therefore, we can check each of the
triplets two, two, five and two, five, five to determine if they can represent the
side lengths of a triangle.

When considering the two, two, five
triplet, we will need to check if the sum of the lengths of the two shorter sides is
greater than the length of the third side. However, two plus two is less than
five. This means that this triplet does
not satisfy the triangle inequality. For the second triplet, we see that
two plus five is greater than five, five plus two is greater than five, and five
plus five is greater than two. Since all three sums are larger
than the length of the third side, this means that the triplet satisfies the
triangle inequality.

We have shown that the only
possibilities for the length of 𝐴𝐶 were two centimeters or five centimeters. Then, we found out that a triangle
with side lengths two centimeters, two centimeters, and five centimeters cannot
exist because the sum of the two shorter side lengths is not greater than the third
side length, whereas the lengths two centimeters, five centimeters, and five
centimeters fulfills the requirements of all three triangle inequalities. Therefore, the third side of the
isosceles triangle 𝐴𝐶 has a length of five centimeters.