Tuesday, August 31, 2021

Triangles in a diagonally segmented trapezium/trapezoid

When a trapezium (quadriletaral with two parallel sides) is segmented diagonally, you end up with 4 triangles as shown below:

These 4 triangles have interesting properties which I shall be proving here, namely that triangles P and Q are similar (same angles, different sizes) and triangles R and S have the same area.

Let's look at just triangles P and Q.

The angles with the same colour are equal for the following reasons: the green angles are opposite angles whilst the red and blue angles are alternative angles.

Now let's look at just triangles Q, R, and S.

In order to show that R and S have the same area, we'll first look at combined triangles Q and R and combined triangles Q and S.

These two combined triangles have the same base and height, which means that they have the same area ($A = \frac{1}{2}bh$). Now triangle Q in both figures is the same triangle, which means that it has the same area. Therefore, removing the area of triangle Q from both shapes will still result in an equal area ($R+Q = S+Q \implies R = S$). So therefore, triangles R and S have the same area.

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