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3.3 – Similar Right Triangles
- 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.
|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!
- 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.
- 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?
- 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
- : Set up the proportion
- : Cross multiply
- : Divide both sides by 10
- : 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
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