Special Right Triangles

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Special Right Triangles
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Objectives

• Use properties of 45 - 45 - 90 triangles
• Use properties of 30 - 60 - 90 triangles

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Side Lengths of Special Right ?s

• Right triangles whose angle measures are 45 -
  45 - 90 or 30 - 60 - 90 are called special
  right triangles. The theorems that describe the
  relationships between the side lengths of each of
  these special right triangles are as follows

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45 - 45 - 90?

• Theorem 7.6
• In a 45- 45- 90 triangle, the length of the
  hypotenuse is v2 times the length of a leg.
• hypotenuse v2 leg

xv2
45
45
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Example 1
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?
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Example 1
Answer Since there are 8 of these triangles in
one square quadrant, the area of one of these
squares is 8(24.5) or 196 mm2.
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Your Turn
Answer 80 mm
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Example 2
Find a.
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Example 2
Rationalize the denominator.
Multiply.
Divide.
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Your Turn
Find b.
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30 - 60 - 90?
Be sure you realize the shorter leg is opposite
the 30? the longer leg is opposite the 60?.
Theorem 7.7

• In a 30- 60- 90 triangle, the length of the
  hypotenuse is twice as long as the shorter leg,
  and the length of the longer leg is v3 times as
  long as the shorter leg.

60
30
xv3
Hypotenuse 2 shorter leg Longer leg v3
shorter leg
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Example 3
Find QR.
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Example 3
Multiply each side by 2.
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Your Turn
Find BC.
Answer BC 8 in.
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Example 4
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Example 4
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Example 4
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Your Turn