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We make a living by what we get, but we make a life by what we
give. -- Winston Churchill
Special Right Triangles
Chapter 8 Section 3
Learning Goal: Use properties of 45°-45 °-90 °, and
30 °-60 °-90 ° Triangles
45°-45°-90° Triangles

x
Special Right Triangle
d 2 = x2 + x 2
Simplify: d2 = 2x2
x
d
√d2 = √ 2x2
d = x√ 2
Three sides of lengths x, x, x√2
What did we
learn about
ratios of sides?
Ratio of a 45°-45°-90°
triangle is:
1 : 1 : √2
45°-45°-90° Triangles

Find the missing side
6
a = 4√2 cm
45°
6√2
45°-45°-90° Triangles
3
8
21
14
Special Right Triangles
WALLPAPER TILING The wallpaper in the figure can
be divided into four equal square quadrants so that
each square contains 8 triangles. What is the area of
one of the squares if the hypotenuse of each 45°–45°–
90° triangle measures
millimeters?
A = 24.5 mm
30°-60°-90° Triangles
Consider an equilateral ∆
a2 = (2x)2 – x2
Simplify: a2 = 4x2 – x2
a2 = 3x2
√a2 = √3x2
2x
2x
a
60°
x
a = x√3
Three sides of lengths x, 2x, x√3
Ratios of sides?
60°
x
Ratio of a 30°-60°-90°
triangle is:
1 : √3 : 2
30°-60°-90° Triangles

Find the missing sides
4
60°
10
5
30°
5√3
8√3
3
30°-60°-90° Triangles
6
8
4√2
Find the Altitude of the Δ
Special Right Triangles
Refer to the figure. Find x and y.
Special Right Triangles
The length of the diagonal of a
square is
cm. Find the
perimeter of the square.
60 cm
Special Right Triangles
The side of an equilateral triangle
measures 21 inches. Find the length
of an altitude of the triangle.
Homework
Special Right Triangles 45-45-90, 30-60-90