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Special Right TrianglesSpecial Right Triangles
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Use properties of 45° - 45° - 90° triangles
Use properties of 30° - 60° - 90° triangles
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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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Theorem 7.6In a 45°- 45°- 90° triangle, the length of the hypotenuse is √2 times the length of a leg.
hypotenuse = √2 • leg
x√245 °
45 °
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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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The length of the hypotenuse of one 45°- 45°- 90° triangle is millimeters. The length of the hypotenuse is times as long as a leg. So, the length of each leg is 7 millimeters.
The area of one of these triangles is
or 24.5 millimeters.
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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WALLPAPER TILING If each 45°- 45°- 90° triangle in the figure has a hypotenuse of millimeters, what is the perimeter of the entire square?
Answer: 80 mm
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Find a.
The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle.
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Multiply.
Divide.
Rationalize the denominator.
Divide each side by
Answer:
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Find b.
Answer:
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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 √3 times as long as the shorter leg.
Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg
x√3
60 °
30 °
Be sure you realize the shorter leg is opposite the 30° & the longer leg is opposite the 60°.
Theorem 7.7
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Find QR.
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is the longer leg, is the shorter leg, and is the hypotenuse.
Multiply each side by 2.
Answer:
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Find BC.
Answer: BC = 8 in.
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COORDINATE GEOMETRY is a 30°-60°-90° triangle with right angle X and as the longer leg. Graph points X(-2, 7) and Y(-7, 7), and locate point W in Quadrant III.
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Graph X and Y. lies on a horizontal gridline of the coordinate plane. Since will be perpendicular to it lies on a vertical gridline. Find the length of
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is the shorter leg. is the longer leg. So, Use XY to find WX.
Point W has the same x-coordinate as X. W is located units below X.
Answer: The coordinates of W are or about
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COORDINATE GEOMETRY is at 30°-60°-90° triangle with right angle R and as the longer leg. Graph points T(3, 3) and R(3, 6) and locate point S in Quadrant III.
Answer: The coordinates of S are or about