Similar Right Triangles
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.
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.
Identify similar triangles
Example 1:
Identify the similar triangles in the diagram.
Solution:
Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation.
ΔQRS ~ ΔPQS ~ Δ PRQ
Example 2:
Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.
Solution:
Draw diagram.
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.
Solution:
Step 1: Draw the three similar triangles
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?
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.
- 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.
- 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.
- 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.
- 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.
- Use the diagram. Find FH.
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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