In today’s geometry lesson, you’re going to learn all about similar right triangles.
Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)
More specifically, you’re going to see how to use the geometric mean to create proportions, which in turn help us solve for missing side lengths.
Let’s get started!
How are right triangles and the geometric mean related?
A right triangle has two acute angles and one 90° angle.The two legs meet at a 90° angle, and the hypotenuse is the side opposite the right angle and is the longest side.
Right Triangle Diagram
The geometric mean of two positive numbers a and b is:
Geometric Mean of Two Numbers
And the geometric mean helps us find the altitude of a right triangle! In fact, the geometric mean, or mean proportionals, appears in two critical theorems on right triangles.
In a right triangle, if the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments, then the length of the altitude is the geometric mean of the lengths of the two segments.
Additionally, the length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg, as ck-12 accurately states.
But what do these theorems really mean?
They help us to create proportions for finding missing side lengths!
Let’s look at an example!
In the figure below, we are being asked to find the altitude, using the geometric mean and the given lengths of two segments:
Using Similar Right Triangles
In the video below, you’ll learn how to deal with harder problems, including how to solve for the three different types of problems:
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