Use Similar Triangles to Solve Problems

Similar triangles are formed when you have two different triangles both sharing the same three angles. This makes their lengths, while different, proportional to one another. This suggests that similar triangles are not congruent figures, meaning that they’re not identical in both length and angles, as shown below.The first video below will show you how to mathematically identify a pair of similar triangles and explain why they are similar using symbols and letters. You’ll discover that to refer to any angle in a triangle (↓), let’s say angle A (∠A) in the one below – relative to B and C – we write ∠BAC or ∠CAB. Notice that the angle in reference is always in the middle of the statement.

The geometry of similar figures is a powerful area of mathematics and can be extend beyond just triangles. Similar triangles can be used to measure the heights of objects that are difficult to get to, such as trees, tall buildings, and cliffs. A demonstration of this is shown in the two examples below:

This relationship explained in question (2) of the video above holds true for all similar figures: The ratio of the areas of two similar figures is equal to the square of the scale factor, k.

Another way to write it is:   A_{\triangle PQR} = k^2 \times A_{\triangle ABC} .

Here’s one last video that deals exclusively with the formula derived above: