Question Video: Triangles Sharing the Same Base and between Parallel Lines Mathematics • 6th Grade

Video Transcript

If the areas of triangle 𝐿𝑁𝐴 and triangle 𝑌𝐴𝐺 are the same, which of the following must be true? Is it (A) 𝑌𝐿 equals 𝑁𝐺? (B) Line segment 𝑌𝐿 is parallel to line segment 𝑁𝐺. Is it (C) 𝐴𝑁 equals 𝐴𝐺? (D) 𝑌𝐺 equals 𝑁𝐿. Or (E) line segment 𝑌𝐺 is parallel to line segment 𝑁𝐿.

Now, be a little bit careful. Each of these lines is drawn on a two-dimensional plane. This is not a 3D shape. It does look a little bit like a square-based pyramid, but it’s certainly not. We’re told that the areas of triangle 𝐿𝑁𝐴 and triangle 𝑌𝐴𝐺 are the same. Let’s highlight these in pink and yellow, respectively. This is triangle 𝐿𝑁𝐴, and we have triangle 𝑌𝐴𝐺 here.

And so let’s recall what we know about the equality of the areas of two triangles. We know that the areas of two triangles will be equal if these triangles share the same base and their vertices lie on a straight line parallel to the base, or if the triangles share the same vertex and have bases of equal length along the same line. And so what we’re going to do to be able to use one of these definitions is split our triangles up.

If we split triangle 𝐿𝑁𝐴 and 𝑌𝐴𝐺 as shown by adding the line segments 𝐷𝑁 and 𝐺𝐸, respectively, then we can begin by considering a shared vertex 𝐴. This corresponds to the second definition. These triangles share the same vertex. And so we’ll be able to say that the area of triangle 𝐴𝐺𝐸 will be equal to the area of triangle 𝐴𝐷𝑁 if their bases are equal length and they lie along the same line. Well, in fact, the dashes on line segment 𝐺𝐸 and 𝐷𝑁 indicate that they’re the same length. And we can quite clearly see from the picture that they do lie along the same line. This must mean that the area of triangle 𝐴𝐺𝐸 is equal to the area of triangle 𝐴𝐷𝑁.

Well, since the areas of our larger triangles are the same, this in turn means that the area of triangle 𝐺𝐸𝑌 must be equal to the area of triangle 𝐷𝑁𝐿. These triangles don’t have a shared vertex. But they do have bases of equal length, which lie on the same line. That’s 𝐺𝐸 and 𝐷𝑁. And so we can use our first definition.

Now, it doesn’t really matter that 𝐺𝐸 and 𝐷𝑁 are different line segments. The simple fact that they lie on the same line and they’re equal in length is good enough as saying that they share the same base. And so we can say that for triangle 𝐺𝐸𝑌 to have an equal area to triangle 𝐷𝑁𝐿, the vertices 𝑌 and 𝐿 — that’s these two — must lie on a line parallel to the base. And so the lines passing through 𝐺𝐸𝐷𝑁 and 𝑌𝐿 must be parallel. If we go to our options, we see that corresponds to option (D). The line segment 𝑌𝐿 must be parallel to the line segment 𝑁𝐺. The answer is (B).