Video Transcript
Given that line 𝐵𝑁 is parallel to line 𝐴𝑀, which of the following has the same area as triangle 𝐴𝐵𝐶? (A) Triangle 𝐿𝐸𝑀, (B) triangle 𝐿𝑀𝑁, (C) 𝐵𝐶𝐸𝐷, (D) 𝐷𝐸𝑀𝐿, or (E) 𝐷𝐹𝑀𝐿.
First of all, let’s go ahead and identify triangle 𝐴𝐵𝐶. That’s this triangle. And we can also mark that line 𝐵𝑁 is parallel to line 𝐴𝑀. There are few other things we should consider, the first being how we find the area of a triangle. The area equals one-half times the height times the base. If we say that the distance from 𝐴 to 𝐶 is the base 𝐵, we notice that this is the same distance from 𝐸 to 𝐹 and from 𝐿 to 𝑁.
We also need to note that the height of a triangle is the perpendicular distance from the base to the opposite vertex. Between these two parallel lines, we have three triangles that all have the same base. And because we know that these two lines are parallel, the distance between the base and the opposite vertex for all three of these triangles will be the same. And so we can say the area of triangle 𝐴𝐵𝐶 is equal to the area of triangle 𝐸𝐷𝐹, which is equal to the area of triangle 𝐿𝑀𝑁. And triangle 𝐿𝑀𝑁 is one of our answer choices.
If we consider triangle 𝐿𝐸𝑀, we see that this triangle does share the same height as triangle 𝐴𝐵𝐶. However, it has a larger base, which means it will have a slightly larger area. And just a note about the three quadrilaterals we have as answer choices, all three of these are trapezoids. And when we find the area of a trapezoid, we take base one, add base two, divide by two, and then multiply by the height.
Now, these trapezoids that are listed do share the height of our triangle 𝐴𝐵𝐶. However, we don’t have enough information to find base one and base two. And therefore, we cannot say that these trapezoids would have the same area as triangle 𝐿𝑀𝑁.
