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.
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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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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Check It Out! Example 1b
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 16.
Rationalize the denominator.
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Example 2 Craft Application
Jana is cutting a square of material for a
tablecloth. The tables diagonal is 36 inches.
She wants the diagonal of the tablecloth to be an
extra 10 inches so it will hang over the edges of
the table. What size square should Jana cut to
make the tablecloth? Round to the nearest inch.
Jana needs a 45-45-90 triangle with a
hypotenuse of 36 10 46 inches.
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Check It Out! Example 2
What if...? Tessas other dog is wearing a square
bandana with a side length of 42 cm. What would
you expect the circumference of the other dogs
neck to be? Round to the nearest centimeter.
Tessa needs a 45-45-90 triangle with a
hypotenuse of 42 cm.
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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.
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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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Check It Out! Example 3a
Find the values of x and y. Give your answers in
simplest radical form.
Hypotenuse 2(shorter leg)
Divide both sides by 2.
y 27
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Check It Out! Example 3b
Find the values of x and y. Give your answers in
simplest radical form.
y 2(5)
Simplify.
y 10
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Check It Out! Example 3c
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.
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Check It Out! Example 3d
Find the values of x and y. Give your answers in
simplest radical form.
Rationalize the denominator.
Hypotenuse 2(shorter leg)
x 2y
Simplify.
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Example 4 Using the 30º-60º-90º Triangle Theorem
An ornamental pin is in the shape of an
equilateral triangle. The length of each side is
6 centimeters. Josh will attach the fastener to
the back along AB. Will the fastener fit if it is
4 centimeters long?
Step 1 The equilateral triangle is divided into
two 30-60-90 triangles.
The height of the triangle is the length of the
longer leg.
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Example 4 Continued
Step 2 Find the length x of the shorter leg.
Hypotenuse 2(shorter leg)
6 2x
3 x
Divide both sides by 2.
Step 3 Find the length h of the longer leg.
The pin is approximately 5.2 centimeters high. So
the fastener will fit.
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Check It Out! Example 4
What if? A manufacturer wants to make a larger
clock with a height of 30 centimeters. What is
the length of each side of the frame? Round to
the nearest tenth.
Step 1 The equilateral triangle is divided into
two 30º-60º-90º triangles.
The height of the triangle is the length of the
longer leg.
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Check It Out! Example 4 Continued
Step 2 Find the length x of the shorter leg.
Rationalize the denominator.
Step 3 Find the length y of the longer leg.
Hypotenuse 2(shorter leg)
y 2x
Simplify.
Each side is approximately 34.6 cm.
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Lesson Quiz Part I
Find the values of the variables. Give your
answers in simplest radical form. 1. 2.
3. 4.
x 10 y 20
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Lesson Quiz Part II
Find the perimeter and area of each figure. Give
your answers in simplest radical form. 5. a
square with diagonal length 20 cm 6. an
equilateral triangle with height 24 in.