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Inscribed Similar Triangles

Division of a right triangle into two similar triangles.

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Inscribed Similar Triangles

Inscribed Similar Triangles Theorem

Remember that if two objects are similar, their corresponding angles are congruent and their sides are proportional in length. The altitude of a right triangle creates similar triangles.

Inscribed Similar Triangles Theorem: If an altitude is drawn from the right angle of any right triangle, then the two triangles formed are similar to the original triangle and all three triangles are similar to each other.

In and :

So, :

This means that all of the corresponding sides are proportional. You can use this fact to find missing lengths in right triangles.

What if you drew a line from the right angle of a right triangle perpendicular to the side that is opposite that angle? How could you determine the length of that line?

Examples

Example 1

Find the value of .

Set up a proportion.

Example 2

Now find the value of in above.

Use the Pythagorean Theorem.

Example 3

Find the value of .

Separate the triangles to find the corresponding sides.

Set up a proportion.

Example 4

Find the value of .

Set up a proportion.

Example 5

Find the values of and .

Separate the triangles. Write a proportion for .

Set up a proportion for . Or, now that you know the value of you can use the Pythagorean Theorem to solve for . Use the method you feel most comfortable with.

Review

Fill in the blanks.

Write the similarity statement for the right triangles in each diagram.

Use the diagram to answer questions 7-10.

1. Write the similarity statement for the three triangles in the diagram.
2. If and , find .
3. Find .
4. Find .

Find the length of the missing variable(s). Simplify all radicals.

1. Fill in the blanks of the proof for the Inscribed Similar Triangles Theorem.

Given: with and is a right angle.

Prove:

Statement Reason
1. 1. Given
2. and are right angles 2.
3. 3.
4. 4. Reflexive PoC
5. 5. AA Similarity Postulate
6. 6.
7. 7.
8. 8.

To see the Review answers, open this PDF file and look for section 8.4.

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Color Highlighted Text Notes

Vocabulary Language: English

Inscribed Similar Triangles Theorem

The Inscribed Similar Triangles Theorem states that if an altitude is drawn from the right angle of any right triangle, then the two triangles formed are similar to the original triangle and all three triangles are similar to each other.

Perpendicular

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.

Proportion

A proportion is an equation that shows two equivalent ratios.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.