Applying Special Right Triangles

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Applying Special Right Triangles
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up For Exercises 1 and 2, find the value of
x. Give your answer in simplest radical form. 1.
2. Simplify each expression. 3. 4.
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Objectives
Justify and apply properties of 45-45-90
triangles. Justify and apply properties of 30-
60- 90 triangles.
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A diagonal of a square divides it into two
congruent isosceles right triangles. Since the
base angles of an isosceles triangle are
congruent, the measure of each acute angle is
45. So another name for an isosceles right
triangle is a 45-45-90 triangle.
45
45
A 45-45-90 triangle is one type of special
right triangle. You can use the Pythagorean
Theorem to find a relationship among the side
lengths of a 45-45-90 triangle.
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The Isosceles Right
50
Why?
a2 b2 c2
l2 l2 h2
2l2 h2
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45
45
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Example 1A Finding Side Lengths in a 45- 45º-
90º Triangle
Find the value of x. Give your answer in simplest
radical form.
By the Triangle Sum Theorem, the measure of the
third angle in the triangle is 45. So it is a
45-45-90 triangle with a leg length of 8.
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Example 1B Finding Side Lengths in a 45º- 45º-
90º Triangle
Find the value of x. Give your answer in simplest
radical form.
The triangle is an isosceles right triangle,
which is a 45-45-90 triangle. The length of
the hypotenuse is 5.
Rationalize the denominator.
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Check It Out! Example 1a
Find the value of x. Give your answer in simplest
radical form.
Simplify.
x 20
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A 30-60-90 triangle is another special right
triangle. You can use an equilateral triangle to
find a relationship between its side lengths.
30
60
60
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Right
51
This can also be proven algebraically, but lets
move on to some problems.
In your reading, notice the corollary to Theorem
51. This corollary involves equilateral
triangles and a familiar formula.
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Example 3A Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your answers in
simplest radical form.
Hypotenuse 2(shorter leg)
22 2x
Divide both sides by 2.
11 x
Substitute 11 for x.
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Example 3B Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your answers in
simplest radical form.
Rationalize the denominator.
Hypotenuse 2(shorter leg).
y 2x
Simplify.
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Example 3C Finding Side Lengths in a 30º-60º-90º
Triangle
Find the values of x and y. Give your answers in
simplest radical form.
Hypotenuse 2(shorter leg)
24 2x
Divide both sides by 2.
12 x
Substitute 12 for x.