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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!

 

GeoA 03.03 - Similar Right Triangles

  • 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}

GeoA 03.03 - Similar Right Triangle Ex1


GeoA 03.03 - Similar Right Triangle Ex2

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

GeoA 03.03 - Similar Right Triangles Ex3

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