Question Video: Finding the Missing Side Length of an Isosceles Triangle Using The Triangle Inequality | Nagwa Question Video: Finding the Missing Side Length of an Isosceles Triangle Using The Triangle Inequality | Nagwa

# Question Video: Finding the Missing Side Length of an Isosceles Triangle Using The Triangle Inequality Mathematics • Second Year of Preparatory School

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If 𝐴𝐵𝐶 is an isosceles triangle with 𝐴𝐵 = 2 cm and 𝐵𝐶 = 5 cm, find 𝐴𝐶.

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### 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.

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