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Question Video: Finding Unknown Side Lengths by Using the Corollaries of Isosceles Triangles Mathematics

In the figure, 𝐴𝐡 = 𝐴𝐢 = 10 cm, 𝐸𝐡 = 𝐸𝐢, and line 𝐴𝐷 ∩ line segment 𝐡𝐢 = 𝐷. If 𝐡𝐢 = 16 cm, what are the lengths of line segment 𝐴𝐷 and line segment 𝐷𝐢 respectively?


Video Transcript

In the figure, 𝐴𝐡 equals 𝐴𝐢 equals 10 centimeters, 𝐸𝐡 equals 𝐸𝐢, and the intersection of line 𝐴𝐷 and line segment 𝐡𝐢 is equal to 𝐷. If 𝐡𝐢 equals 16 centimeters, what are the lengths of line segment 𝐴𝐷 and line segment 𝐷𝐢, respectively?

We can begin this question by looking at the information that we are given. And since there are two congruent line segments, then we can add the information to the diagram that 𝐴𝐡 is also 10 centimeters. We can note that there are two triangles that have special properties. They are triangle 𝐴𝐡𝐢 and triangle 𝐸𝐡𝐢. They are both isosceles triangles as they each have a pair of congruent sides.

Now, we need to find the lengths of line segments 𝐴𝐷 and 𝐷𝐢. And it doesn’t seem as though we have enough information, only that we have determined that we have two isosceles triangles. However, there might be some corollaries of the isosceles triangle theorems that could help us: this one in particular, that the bisector of the vertex angle of an isosceles triangle is a perpendicular bisector of the base.

But in order to use this corollary, we need to have the bisector of the vertex angle. So, if we take triangle 𝐴𝐡𝐢, do we have the vertex angle of 𝐢𝐴𝐡 bisected? That is, is the measure of angle 𝐢𝐴𝐷 equal to the measure of angle 𝐡𝐴𝐷? Now, of course, we know that it isn’t enough just to guess or say that they look the same. We must prove that they are.

Let’s take a detour from the main question and consider the triangles 𝐴𝐸𝐢 and 𝐴𝐸𝐡. In these triangles, we know that we have a pair of congruent sides, since 𝐴𝐡 equals 𝐴𝐢. And there is another pair of congruent sides, since we were given that 𝐸𝐡 equals 𝐸𝐢. Furthermore, the remaining side 𝐴𝐸 is common to both triangles, so it is congruent in each triangle.

These triangles have three pairs of sides congruent. And therefore, we can write that triangles 𝐴𝐸𝐢 and 𝐴𝐸𝐡 are congruent by the SSS, or side-side-side, congruency criterion. That means that our angles 𝐢𝐴𝐷 and 𝐡𝐴𝐷 do have equal angle measures, as these would be the corresponding angles 𝐢𝐴𝐸 and 𝐡𝐴𝐸 in the congruent triangles.

And importantly, we have proved that in the largest triangle 𝐴𝐡𝐢, the vertex angle is bisected by the line segment 𝐴𝐷. So let’s clear some space and see what this corollary will now help us prove.

The corollary tells us that the bisector of the vertex angle is a perpendicular bisector of the base. That means that the measure of angle 𝐴𝐷𝐢 is 90 degrees and that the line segments 𝐷𝐢 and 𝐷𝐡 are congruent. We were given that the length of 𝐡𝐢 is 16 centimeters. And as this line segment is bisected, then both 𝐷𝐢 and 𝐷𝐡 would have a length of eight centimeters. And that means that we have found the length of one of the required line segments, 𝐷𝐢.

But we still need to calculate the length of line segment 𝐴𝐷. We might notice that this line segment 𝐴𝐷 appears in triangles 𝐴𝐷𝐢 and 𝐴𝐷𝐡, both of which are right triangles. If we take triangle 𝐴𝐷𝐢, we know the side lengths of two sides of this right triangle, and we need to find the length of the third side. So, we can apply the Pythagorean theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides. Often this is written as π‘Ž squared plus 𝑏 squared equals 𝑐 squared, where π‘Ž and 𝑏 are the two shorter sides and 𝑐 is the hypotenuse.

The two shorter sides are 𝐴𝐷 and 𝐷𝐢, which has a length of eight centimeters, and the hypotenuse is 𝐴𝐢, which has a length of 10 centimeters. We may already notice that we have a Pythagorean triple, but if not, we can continue the process. We can calculate the squares as 64 and 100 and then subtract 64 from both sides, leaving us with 𝐴𝐷 squared equals 36. Finally, we must remember to take the square root of both sides. And as 𝐴𝐷 is a length, then we know that we must have a positive value for it. So 𝐴𝐷 is six centimeters.

We can therefore give the answers for both line segments. 𝐴𝐷 is six centimeters, and 𝐷𝐢 is eight centimeters.

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