Video Transcript
If the area of triangle 𝐷𝐸𝐶 is equal to 6.99 centimeters squared, find the area of triangle 𝐴𝐵𝐶.
In this question, we are given a figure and the area of a triangle on this figure. We want to use this information to determine the area of another triangle in the figure. To do this, we can start by highlighting triangle 𝐷𝐸𝐶 on the diagram and adding that its area is 6.99 centimeters squared. We cannot directly compare the area of this triangle to triangle 𝐴𝐵𝐶. However, we can note that side 𝐸𝐶 has the same length as side 𝐴𝐸. We can then compare the area of triangle 𝐸𝐶𝐷 to 𝐴𝐸𝐷. We see that both triangles have a side of the same length, so we can choose this to be the base of the triangle.
The area of a triangle is given by half the length of its base times the perpendicular height between the base and the other vertex. We can see that both triangles 𝐴𝐸𝐷 and 𝐸𝐶𝐷 share the same perpendicular height, ℎ. Since these triangles have the same length base and same height, they have the same area. So triangle 𝐴𝐸𝐷 has area 6.99 centimeters squared.
It is worth noting that we could have also shown that these triangles have the same area by using the following result, although the method we used outlines the proof of the result. If we have any two triangles that share congruent bases on the same straight line and they share a common vertex on the opposite side of the base, then they must have the same area.
We can apply this result by noting that the missing triangle whose area we want to find, that is, triangle 𝐴𝐷𝐵, we see that it has a congruent base with triangle 𝐴𝐶𝐷. And they share a common vertex opposite the base at 𝐴. Therefore, the area of triangle 𝐴𝐷𝐵 is equal to the area of triangle 𝐴𝐶𝐷. We can find the area of this triangle by adding the areas of the smaller triangles that combine to make it, that is, 6.99 plus 6.99. Adding these areas together gives us that the area of triangle 𝐴𝐷𝐵 is 13.98 centimeters squared. Finally, we can add the areas of these triangles together to obtain that the area of triangle 𝐴𝐵𝐶 is 27.96 centimeters squared.
