Question Video: Identifying the Triangle Having the Same Area as Another Given Triangle Using the Relations between Them and between Parallel Lines

Video Transcript

Which triangle has the same area as triangle 𝑁𝑀𝐾?

Triangle 𝑁𝑀𝐾 is found here. The area of a triangle is equal to one-half times the base times the height. So let’s look at the base of each triangle. The base will run on the parallel lines, because the base doesn’t necessarily mean the bottom. We can choose any side, but we have to be careful and make sure that the height is perpendicular to it.

So here is the base of 𝑀𝑁, here is the base of 𝐶𝑍, and here’s the base of 𝑂𝐷. Notice that all three of the bases are congruent. So right now, any of the three triangles could work. So what about the height? The heights that would be the same as triangle 𝑁𝑀𝐾 would then have the same area as triangle 𝑁𝑀𝐾, because they already have the same base.

The height of triangle 𝑁𝑀𝐾 would be here; it’s the distance between the parallel lines. The height of triangle 𝐶𝑍𝐻 is found here; it’s the distance between the parallel lines. And it’s the exact same for the last triangle. So they all have the same base, and they all have the same height. That means their areas would all be equal.

So which triangle has the same area as triangle 𝑁𝑀𝐾? It would be either triangle 𝐶𝑍𝐻 or triangle 𝑂𝐷𝑋, because if two triangles with equal base links lie between two parallel straight lines, then the triangles are equal in area.