Video Transcript
Which of the following has the same area as triangle 𝐷𝐸𝐵? (A) Triangle 𝐸𝐷𝐶, (B) triangle 𝐸𝐹𝐶, (C) triangle 𝐹𝐵𝐶, (D) 𝐴𝐷𝐹𝐸, or (E) 𝐷𝐵𝐶𝐸.
First, let’s identify triangle 𝐷𝐸𝐵. This is triangle 𝐷𝐸𝐵 in our figure. We know to find the area of the triangle, we take one-half the height times the base. And in triangle 𝐷𝐸𝐵, we could let 𝐷𝐸 be the base. The height of a triangle is the perpendicular distance from the base to the opposite vertex. Our opposite vertex will be 𝐵, which means the height will be the perpendicular distance from line 𝐷𝐸 to line 𝐵𝐶.
To find a triangle that has the same area, we need a triangle with the same base and the same height. Since we don’t know any of the distances here, the only triangle that would have the same base as triangle 𝐷𝐸𝐵 would be a triangle that shares the base 𝐷𝐸. For example, triangle 𝐷𝐸𝐴 shares the base 𝐷𝐸, as does triangle 𝐷𝐸𝐶. But to have a triangle with the same area, we don’t just need the same base; we need the same height. We have no way of knowing if the perpendicular distance from the base 𝐷𝐸 to the vertex 𝐴 is the same.
However, when we look at triangle 𝐷𝐸𝐶, we recognize that the perpendicular distance for this triangle, its height, falls between the same two parallel lines as triangle 𝐷𝐸𝐵. Because line 𝐷𝐸 is parallel to line 𝐵𝐶, these two perpendicular heights are equal, which shows us that our initial triangle 𝐷𝐸𝐵 will have the same area as this triangle, triangle 𝐷𝐸𝐶. When listing out triangles, we can list them in a few different orders. Triangle 𝐸𝐷𝐶 is the same triangle as triangle 𝐷𝐸𝐶, and that is option (A).
