Similar Right Triangles

Definition

Triangles are similar if there is a corresponding between their vertices such that corresponding angles are congruent and the measures of the corresponding sides are proportional. In other words, triangles that are the same shape but different sizes are similar.

two triangles of the same shape, but with different sizes and rotated to different angles

Go to Explore! Similar Triangles to find out more.

Similar Right Triangles

If the altitude is drawn from the right angle of a right triangle to the hypotenuse, then the two right triangles formed are similar to the given right triangle and to each other.

A triangle with angles A, B, and C. Angle C drops an altitude to create point D.

In the given triangle

△ADC ~ △ACB

△BDC ~ △ACB

△ADC ~ △BDC

Since △ADC ~ △ACB, △BDC ~ △ACB, △ADC ~ △BDC

Thus, by substitution, all three triangles are similar.

Labeling Similar Right Triangles

We are going to label the triangle in a way that will help us express the relationship between sides. Let's break the two smaller triangles out.

The previously pictured triangle is split into three triangles: the original triangle and the two triangles formed by dropping the altitude

Now, let's label each side. We label side opposite angles with a lowercase letter matching the angle.

There are still a few sides that are unlabeled. Let's name them x, y, and h. Notice that x and y are segments of c, so x + y = c.

△ACB ~ △ADC ~ △BDC

Notice the hypotenuse, short leg, and long leg in each triangle.

Similar Right Triangles

A triangle's hypotenuse A to B is split by altitude C to D. The altitude splits the hypotenuse into two smaller triangles. The first triangle has side y, which is part of the hypotenuse, side b, and the altitude. The second triangle has side x, which is part of the hypotenuse, side a, and the altitude.

Since we know that the three triangles formed by the altitude are all similar, we can express the relationships between sides as a set ratios. Fill in the ratios below.

long over short equals b over a equals y over blank equals blank over x

Answer: h, h; Look at triangle ADC. What is its short side? Now, look at triangle CDB. What is its long side?

hypotenuse over long equals c over b equals b over y equals a over blank

Answer: h

hypotenuse over short equals c over a equals b over blank equals a over x

Answer: h

Geometric Mean

Thus given a right triangle with an altitude drawn from the right angle there is a special relationship with the altitude and the segment partitions of they hypotenuse.

A triangle's hypotenuse A to B is split by altitude C to D. The altitude splits the hypotenuse into two smaller triangles. The first triangle has side y, which is part of the hypotenuse, side b, and the altitude. The second triangle has side x, which is part of the hypotenuse, side a, and the altitude.

Geometric Mean Theorem:

y over h equals h over x

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