This is a really interesting and surprising characteristic of parallelograms, that the four triangles formed from the intersection of their diagonals have the same area! I'm glad you asked about it.

For your first question, actually, there are three ways of thinking of a base and height for a given triangle. This is because there are three sides of a triangle! In the diagram below, I have drawn the three different heights for the yellow triangle, as well as the three different heights for the purple triangle. I tried to color-code the heights with their matching bases.

M2W1D4-ch-part-1-solution-bed-abe-smaller.png

The idea is that a height must be perpendicular to the base. That means an altitude might have to lie outside the triangle, which is true for obtuse triangles.

Now, for the four triangles formed from the diagonals of the parallelogram, let's look at this picture:

M2W1D4-ch-why-same-area.png

The purple triangle and yellow triangles have the same altitude, which is labeled as "h." We are choosing here the red line to be the base of the purple triangle, and the green line to be the base of the yellow triangle. Note that the line marked with an \(h\) is indeed perpendicular to both of these base. Now, the exciting part is that, due to the symmetry of the parallelogram, the diagonals cut each other in half, so the red line is the same length as the green line. This is why the area of the purple triangle is equal to the area of the yellow triangle.

The other two triangles which were not colored in the diagram are just flipped versions of these two, so the same applies to them, and any such triangles drawn from any parallelogram's diagonals.

Happy Learning!

The Daily Challenge Team

And here's a quick guide on how to type math symbols into your posts!

Are you confused about the context of the number 7 in this question? My apologies; it's hard to figure out where this came from. The 7 pops out at 1:38 in the video and is the third length of the triangle with sides of 3, 5 and 7 (with a \(120^{\circ}\) angle in between):

M2W1D4-screenshot-3-5-7-triangle.png

The explanation for why the third side of this triangle is 7 can be found around timestamp 8:00 in the Day 1: Your Turn Explanation video..

Does this answer your question? Please let us know; we're more than happy to help! 🙂 🙂

Happy Learning!

@debbie said in Why:

The Daily Challenge Team