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Similar Right Triangles

Sep 13, 2022
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Key Concepts

  • Identify similar triangles

Right angle

the angle bounded by two lines perpendicular to each other: an angle of 90° or ¹/₂ π radians. 

Right angle 1

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

Right angle 2
Right angle 3
Right angle 4

Identify similar triangles 

Example 1: 

Identify the similar triangles in the diagram. 

Example 1: 

Solution: 

parallel

Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. 

Example 1: solution

ΔQRS  ~ ΔPQS ~ Δ PRQ 

Example 2: 

Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 

Example 2: 

Solution: 

parallel

Draw diagram. 

Example 2: solution

x/23 = 12.8 / 26.6

26.6 (x) = 294.4 

x = 11.1 ft 

Example 3: 

Find the value of y. Write your answer in the simplest radical form. 

Example 3: 

Solution: 

Step 1: Draw the three similar triangles 

Example 3: solution
Example 3: solution

Step 2: Write a proportion.  

6/x = x/2 (Substitute) 

12 = x2 (Cross product property) 

√12 = x (Take the positive square root of each side) 

2√3= x (Simplify) 

Example 4: 

A 30 ft building casts a shadow that is 75 ft long. A nearby tree casts a shadow that is 35 ft long. How tall is the tree?  

Example 4: 

Solution: 

𝟑𝟎/𝟕𝟓  𝒉/𝟑𝟓          (Corresponding sides of similar figures are proportional) 

75h = 1050   (Find the cross products)  

𝟕𝟓 / 𝒉𝟕𝟓 = 𝟏𝟎𝟓𝟎/𝟕𝟓    (divides both sides by 75) 

 h = 14 

The height of the tree is 14 feet. 

Exercise

  • Identify similar triangles. Then find the value of x.
Identify similar triangles. Then find the value of x.
  • Charmin is 5.5 feet tall. How far from the wall in the image below would she have to stand in order to measure his height?
Charmin is 5.5 feet tall. How far from the wall in the image below would she have to stand in order to measure his height?
  • Identifying similar triangles: Identify three similar right triangles in the given diagram.
Identifying similar triangles: Identify three similar right triangles in the given diagram.
  • Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.
Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.
  • Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion.          
Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion.    
  • Find the value(s) of the variable(s).
Find the value(s) of the variable(s).
  • Using theorems: Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.
Using theorems: Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.
  • Describe and correct the error in writing a proportion for the given diagram.
Describe and correct the error in writing a proportion for the given diagram.
  • Finding lengths: Use the Geometric Mean Theorems to find AC and BD.
Finding lengths: Use the Geometric Mean Theorems to find AC and BD.
  • Use the diagram. Find FH.
Use the diagram. Find FH.

Concept Map

Concept Map

What have we learned

  • Identify similar triangles
  • Understand how to find the length of the altitude to the hypotenuse
  • Understand  geometric mean
  • Simplest radical form.
  • Understand how to find a height using indirect measurement.

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