### Video Transcript

If the area of triangle π·πΉπΆ equals 93.8 centimeters squared and π΄πΉ equals πΉπΈ, find the area of triangle π΄π΅πΈ.

First, letβs highlight triangle π·πΉπΆ. We know that the area of this space is 93.8 centimeters squared. We also know that the line πΉπ΄ and the line πΈπΉ are equal in length. And weβre trying to find the area of triangle π΄π΅πΈ, this space. However, itβs going to take a few steps for us to figure this out. So letβs go back to what we were originally given.

Before we can find the area of triangle π΄π΅πΈ, letβs try to find other areas we can know inside this triangle. We know that if two triangles share the same base and their vertices lie on a straight line parallel to their base, then the two triangles are equal in area. If we consider the larger triangle, triangle π΄π·πΆ, and the triangle π·π΄π΅, these two triangles have a shared base. And they have their vertices that lie on a straight line that is parallel to their base, which means the area of triangle π΄π·πΆ is equal to the area of triangle π·π΄π΅.

Now both of these triangles have the smaller triangle π΄π·πΉ inside of them. If we take the area of triangle π΄π·πΆ and subtract this smaller triangle, the area of triangle π΄π·πΉ, it would be equal to this space in orange, triangle π·πΉπΆ. And if we take away the area of triangle π΄π·πΉ, from the pink triangle π·π΄π΅, it will be equal to the area in the pink, the area of triangle π΄πΉπ΅. And we can say that these two areas are equal to each other. And since we know that the area of triangle π·πΉπΆ equals 93.8 centimeters squared, we can say that the area of triangle π΄πΉπ΅ is also equal to 93.8 centimeters squared.

Now that weβve established that these two triangles have equal area, how does that help us? And what can that do so that weβll be able to find the area of triangle π΄π΅πΈ? We need another rule about triangles and their areas. We also know that if two triangles share the same vertex and have bases of equal length along the same line, then the heights of the triangles must be equal and the areas of the two triangles must be equal.

Remember that weβre looking for triangle π΄π΅πΈ, this triangle. And inside triangle π΄π΅πΈ, there are two smaller triangles, the triangle with its area highlighted in pink, triangle π΄πΉπ΅, and the smaller triangle πΈπΉπ΅. We noticed that both of these triangles share a vertex π΅. And they have bases of equal length along the same line, which tells us that the height of both of these triangles must be equal and their areas must also be equal.

The area of triangle π΄πΉπ΅ equals the area of triangle πΈπΉπ΅. We can say that the area weβre interested in, triangle π΄π΅πΈ, must be equal to the area of triangle π΄πΉπ΅ plus the area of triangle πΈπΉπ΅, both of which equal 93.8 centimeters squared. 93.8 plus 93.8 equals 187.6 centimeters squared. Based on these two rules, weβve shown that triangle π΄π΅πΈ has an area of 187.6 centimeters squared.