### Video Transcript

In the figure, π΄π΅ equals π΄πΆ equals 10 centimeters, πΈπ΅ equals πΈπΆ, and the intersection of line π΄π· and line segment π΅πΆ is equal to π·. If π΅πΆ equals 16 centimeters, what are the lengths of line segment π΄π· and line segment π·πΆ, 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 π΄π΅ is also 10 centimeters. We can note that there are two triangles that have special properties. They are triangle π΄π΅πΆ and triangle πΈπ΅πΆ. They are both isosceles triangles as they each have a pair of congruent sides.

Now, we need to find the lengths of line segments π΄π· and π·πΆ. And it doesnβt 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 π΄π΅πΆ, do we have the vertex angle of πΆπ΄π΅ bisected? That is, is the measure of angle πΆπ΄π· equal to the measure of angle π΅π΄π·? Now, of course, we know that it isnβt enough just to guess or say that they look the same. We must prove that they are.

Letβs take a detour from the main question and consider the triangles π΄πΈπΆ and π΄πΈπ΅. In these triangles, we know that we have a pair of congruent sides, since π΄π΅ equals π΄πΆ. And there is another pair of congruent sides, since we were given that πΈπ΅ equals πΈπΆ. Furthermore, the remaining side π΄πΈ 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 π΄πΈπΆ and π΄πΈπ΅ are congruent by the SSS, or side-side-side, congruency criterion. That means that our angles πΆπ΄π· and π΅π΄π· do have equal angle measures, as these would be the corresponding angles πΆπ΄πΈ and π΅π΄πΈ in the congruent triangles.

And importantly, we have proved that in the largest triangle π΄π΅πΆ, the vertex angle is bisected by the line segment π΄π·. So letβs 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 π΄π·πΆ is 90 degrees and that the line segments π·πΆ and π·π΅ are congruent. We were given that the length of π΅πΆ is 16 centimeters. And as this line segment is bisected, then both π·πΆ and π·π΅ would have a length of eight centimeters. And that means that we have found the length of one of the required line segments, π·πΆ.

But we still need to calculate the length of line segment π΄π·. We might notice that this line segment π΄π· appears in triangles π΄π·πΆ and π΄π·π΅, both of which are right triangles. If we take triangle π΄π·πΆ, 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 squared plus π squared equals π squared, where π and π are the two shorter sides and π is the hypotenuse.

The two shorter sides are π΄π· and π·πΆ, which has a length of eight centimeters, and the hypotenuse is π΄πΆ, 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 π΄π· squared equals 36. Finally, we must remember to take the square root of both sides. And as π΄π· is a length, then we know that we must have a positive value for it. So π΄π· is six centimeters.

We can therefore give the answers for both line segments. π΄π· is six centimeters, and π·πΆ is eight centimeters.