# 3.3 – Similar Right Triangles

## Objectives

• Determine if right triangles are similar when given only one acute angle measure in each triangle.
• Prove that if an altitude is drawn from the right angle vertex of a right triangle to its hypotenuse, then three similar triangles are formed.
• Calculate the missing sides of similar right triangles using proportions.
• Solve real-world problems by using the properties of similar right triangles.

## Notes

Similar Right Triangles
• Similar right triangles have the same shape, but not always the same size.
• Their corresponding angles are congruent.
• Their corresponding sides are proportional.
• If two triangles are similar, all of their angles are congruent.

• Transitive Property (like a Syllogism)
• If a = b and b = c, then a = c.
• If a ~ b and b ~ c, then a ~ c.

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Right Triangle Similarity Theorem
• You have a right triangle.
• You draw an altitude from its right angle to its hypotenuse.
• This forms two smaller right triangles.
• Both smaller triangles are similar to the original!

• Proof

• How to Use Proportions to Find a Side Length
(A proportion is an equation that shows two ratios are equal).

• Step 1: Write the proportion with side names.
• Step 2: Replace the names with numbers (side lengths you know) and a variable (unknown side length).
• Step 3: Cross multiply.
• Step 4: Solve the equation for the variable.

$\frac{AB}{AC}=\frac{BD}{AD}$

Examples
• Distance from a Tree (Example)
• Leslie places a mirror on the ground 63 feet from the base of a tree. She walks backwards until she can see the top of the tree in the middle of the mirror.  At that point, Val’s eyes are 5 feet above the ground and she is 9 feet from the image in the mirror.  What is the height of the tree?
• Steps
• Step 1: Draw the diagram (like the one above, where the triangles are facing each other)
• Step 2: Label the side lengths
• Step 3: Set up the proportions
• Step 4: Cross multiply
• Step 5: Simplify & solve for x
• Solve
• $\frac{Leslie-from-Tree}{Val-from-Tree}=\frac{Height-of-the-Tree}{Height-of-Val}$
• $\frac{70}{10}=\frac{x}{5}$: Set up the proportion
• $70(5)=10x$: Cross multiply
• $350=10x$: Divide both sides by 10
• $35=x$: The tree is 35 feet tall.
• Two right triangles are can be proven using the AA Similarity Postulate
• The sum of all three angles of a triangle add up to 180.
• Similar right triangles will both have a congruent 90 degree angle.
• You can find the missing angle of each triangle below using this postulate.
• Both triangles have a 90, 72, and 18 degree angles