In the figure, 𝐴𝐵 equals 𝐴𝐶 equals 10 centimeters, 𝐸𝐵 equals 𝐸𝐶, and the intersection of line 𝐴𝐷 and line segment 𝐵𝐶 is equal to 𝐷. If 𝐵𝐶 equals 16 centimeters, what are the lengths of line segment 𝐴𝐷 and line segment 𝐷𝐶, 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 𝐴𝐵 is also 10 centimeters. We can note that there are two triangles that have special properties. They are triangle 𝐴𝐵𝐶 and triangle 𝐸𝐵𝐶. 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 𝐷𝐶. 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 𝐴𝐵𝐶, do we have the vertex angle of 𝐶𝐴𝐵 bisected? That is, is the measure of angle 𝐶𝐴𝐷 equal to the measure of angle 𝐵𝐴𝐷? 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 𝐴𝐸𝐶 and 𝐴𝐸𝐵. In these triangles, we know that we have a pair of congruent sides, since 𝐴𝐵 equals 𝐴𝐶. And there is another pair of congruent sides, since we were given that 𝐸𝐵 equals 𝐸𝐶. 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 𝐴𝐸𝐶 and 𝐴𝐸𝐵 are congruent by the SSS, or side-side-side, congruency criterion. That means that our angles 𝐶𝐴𝐷 and 𝐵𝐴𝐷 do have equal angle measures, as these would be the corresponding angles 𝐶𝐴𝐸 and 𝐵𝐴𝐸 in the congruent triangles.
And importantly, we have proved that in the largest triangle 𝐴𝐵𝐶, 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 𝐴𝐷𝐶 is 90 degrees and that the line segments 𝐷𝐶 and 𝐷𝐵 are congruent. We were given that the length of 𝐵𝐶 is 16 centimeters. And as this line segment is bisected, then both 𝐷𝐶 and 𝐷𝐵 would have a length of eight centimeters. And that means that we have found the length of one of the required line segments, 𝐷𝐶.
But we still need to calculate the length of line segment 𝐴𝐷. We might notice that this line segment 𝐴𝐷 appears in triangles 𝐴𝐷𝐶 and 𝐴𝐷𝐵, both of which are right triangles. If we take triangle 𝐴𝐷𝐶, 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 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏 are the two shorter sides and 𝑐 is the hypotenuse.
The two shorter sides are 𝐴𝐷 and 𝐷𝐶, which has a length of eight centimeters, and the hypotenuse is 𝐴𝐶, 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 𝐷𝐶 is eight centimeters.