Download 10.2 45 -45 -90 Triangles

Page 1 of 6
10.2
Goal
Find the side lengths of
458-458-908 triangles.
Key Words
458-458-908 Triangles
Geo-Activity
Exploring an Isosceles Right Triangle
1 Fold a large piece of paper so the top lines up with one side.
●
2 Measure the
●
• 458-458-908 triangle
• isosceles triangle p. 173
3 Measure the
●
angles of the
triangle formed.
legs of the
triangle.
• leg of a right triangle
p. 192
• hypotenuse p. 192
4 Use the Pythagorean Theorem to predict the length of the hypotenuse.
●
5 Measure the hypotenuse to verify your answer in Step 4.
●
Student Help
LOOK BACK
To review the
Pythagorean Theorem,
see p. 192.
A right triangle with angle measures of 458, 458, and 908 is called a
458-458-908 triangle . You can use the Pythagorean Theorem to find
the length of the hypotenuse of any 458-458-908 triangle.
2
2Ï·
2
2
3Ï·
3
2
3
Ï2w2w1
ww22w 5 Ï4ww
1w
4
Ï3w2w1
ww
3 2w 5 Ï9
ww
1w
9
2
4Ï·
4
4
Ï4w2w1
ww42w 5 Ï1w6ww
1w6
1w
5 Ï4
wwpw
2
5 Ï9
wwpw
2
5 Ï1
w6
wwpw
2
5 Ï4
w p Ï2
w
5 Ï9
w p Ï2
w
5 Ï1
w6
w p Ï2
w
5 2Ï2
w
5 3Ï2
w
5 4Ï2
w
THEOREM 10.1
458-458-908 Triangle Theorem
Words
In a 458-458-908 triangle, the length of the
hypotenuse is the length of a leg times Ï2
w.
Symbols hypotenuse 5 leg p Ï2w
542
Chapter 10
Right Triangles and Trigonometry
x
458
x Ï2
458
x
Page 2 of 6
IStudent Help
EXAMPLE
1
Find Hypotenuse Length
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
Find the length x of the hypotenuse in the
458-458-908 triangle shown at the right.
3
3
458
x
Solution
By the 458-458-908 Triangle Theorem, the length of the
hypotenuse is the length of a leg times Ï2
w.
w
hypotenuse 5 leg p Ï2
5 3 p Ï2
w
ANSWER
458-458-908 Triangle Theorem
Substitute.
© The length of the hypotenuse is 3Ï2
w.
EXAMPLE
2
Find Leg Length
Find the length x of each leg in the
458-458-908 triangle shown at the right.
7Ï·
2
Solution
READING TIP
By the 458-458-908 Triangle Theorem, the length of the
hypotenuse is the length of a leg times Ï2
w.
x
hypotenuse 5 leg p Ï2
w
458-458-908 Triangle Theorem
7Ï2
w 5 xÏ2
w
Substitute.
7Ï2
w
w
x Ï2
}5}
Ï2
w
Ï2
w
Divide each side by Ï2
w.
75x
ANSWER
x
458
Student Help
The expression x Ï2w is
equivalent to Ï2wx .
458
Simplify.
© The length of each leg is 7.
Find Hypotenuse and Leg Lengths
Find the value of x.
1.
2.
x
458
5
458
3.
x
458
458 4
4.
x
458
458
6Ï·
2
458
458
x
3Ï·
2
10.2
458-45 8-90 8 Triangles
543
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EXAMPLE
3
Identify 458-458-908 Triangles
Determine whether there is enough information to
conclude that the triangle is a 458-458-908
triangle. Explain your reasoning.
x8
x8
Solution
By the Triangle Sum Theorem, x81 x8 1 908 5 1808.
So, 2x8 5 908, and x 5 45.
ANSWER
© Since the measure of each acute angle is 458, the triangle is
a 458-458-908 triangle.
Example 3 shows that whenever a right triangle has congruent acute
angles, it is a 458-458-908 triangle.
Student Help
EXAMPLE
LOOK BACK
4
Find Leg Length
Show that the triangle is a 458-458-908 triangle.
Then find the value of x.
To review the Base
Angles Theorem,
see p. 185.
5
x
x
Solution
The triangle is an isosceles right triangle. By the Base Angles Theorem,
its acute angles are congruent. From the result of Example 3, this
triangle must be a 458-458-908 triangle.
You can use the 458-458-908 Triangle Theorem to find the value of x.
hypotenuse 5 leg p Ï2
w
5 5 xÏ2
w
458-458-908 Triangle Theorem
Substitute.
5
xÏ2
w
}5}
Ï2
w
Ï2
w
Divide each side by Ï2
w.
5
}5x
Ï2
w
Simplify.
3.5 ≈ x
Use a calculator to approximate.
Find Leg Lengths
Show that the triangle is a 458-458-908 triangle. Then find the value
of x. Round your answer to the nearest tenth.
5.
6.
x
12
x
x
8
544
Chapter 10
Right Triangles and Trigonometry
Page 4 of 6
10.2 Exercises
Guided Practice
Vocabulary Check
1. How many congruent sides does an isosceles right triangle have?
2. How many congruent angles does an isosceles right triangle have?
What are the measures of the three angles?
Skill Check
Find the value of x in the 458-458-908 triangle. Write your answer in
radical form.
3.
4.
x
6
x
5.
2
2Ï·
x
Ï·
6
Ï·6
458
x
Practice and Applications
Extra Practice
See p. 693.
Finding Hypotenuse Lengths Find the length of the hypotenuse in
the 458-458-908 triangle. Write your answer in radical form.
6.
7.
2
x
458
8.
x
458
458
7
7
8
458
2
9.
10.
458
x
11.
Ï·
3
458
Ï·
5
Ï·
3
458
x
458
Ï·
5
8
458
458
x
Ï·
10·
458
x
458 Ï·10·
Finding Leg Lengths Find the length of a leg in the 458-458-908
triangle.
12.
13.
x
14.
10Ï·
2
4Ï·
2
x
x
x
2
5Ï·
x
x
Homework Help
15.
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 6–11, 18
Exs. 12–17
Exs. 19–27
Exs. 22–27
x
x
Ï·
2
16.
17.
2
14Ï·
x
8Ï·
2
x
10.2
x
458
458-45 8-90 8 Triangles
458
x
545
Page 5 of 6
18. Jewelry Use a calculator to
find the length x of the earring
shown at the right. Round your
answer to the nearest tenth.
1.4 cm
x
1.4 cm
You be the Judge
Determine whether there is enough
information to conclude that the triangle is a 458-458-908 triangle.
Explain your reasoning.
19.
20.
21.
458
Finding Leg Lengths Show that the triangle is a 458-458-908 triangle.
Then find the value of each variable. Round to the nearest tenth.
22.
23.
4
x
24.
32
x
9
458
x
x
25.
26.
8
x8
458
x8
y
27.
20
x
x
35
458
Technology In Exercises 28–30, use geometry software.
&* and construct its midpoint, C.
1 Draw AB
●
&*.
2 Construct the perpendicular bisector of AB
●
3 Construct point D on the bisector and
●
D
&* and DB
&*.
construct AD
4 Measure aADB. Drag point D until
●
A
C
maADB 5 908.
28. Name three 458-458-908 triangles. Explain
how you know they are 458-458-908 triangles.
&*, CB
&*, and CD
&*. What do you notice? Explain.
29. Measure AC
&* and DB
&*. Check your answer by
30. Predict the measures of AD
measuring the segments.
31. Error Analysis A student labels
a 458-458-908 triangle as shown.
Explain and correct the error.
Ï·
5
5Ï·2
Ï·
5
546
Chapter 10
Right Triangles and Trigonometry
B
Page 6 of 6
Quilt Design The quilt design in the photo is based on the pattern in
the diagram below. Use the diagram in Exercises 32 and 33.
1
1
1
1
s
t
u
r
1
1
v
w
1
“Wheel of Theodorus,”
by Diana Venters
32. Working from left to right, use the Pythagorean Theorem in each
right triangle to find the values of r, s, t, u, v, and w.
33. Identify any 458-458-908 triangles in the figure.
Standardized Test
Practice
34. Multi-Step Problem Use the triangle shown below.
a. Find the value of x. Then find
A
maA, maB, and maC.
b. Find the values of b and c.
b
3x8
c
c. Use the Pythagorean Theorem or
the 458-458-908 Triangle Theorem
to justify your answers in part (b).
Mixed Review
6x8
C
3x8
12
B
Classifying Triangles Classify the triangle as acute, right, or obtuse.
(Lesson 4.5)
35.
26
36.
10
9
24
40
37.
16
41
9
14
Simplifying Radicals Simplify the radical expression. (Lesson 10.1)
Algebra Skills
38. Ï2
w4
w
39. Ï6
w3
w
40. Ï5
w2
w
41. Ï6
w4
w
42. Ï8
w0
w
43. Ï1
w9
w6
w
44. Ï2
w5
w0
w
45. Ï1
w1
w7
w
Writing Fractions as Decimals Write the fraction as a decimal.
For repeating decimals, also round to the nearest hundredth.
(Skills Review, p. 657)
3
5
48. }}
3
20
52. }}
9
10
47. }}
4
9
51. }}
46. }}
50. }}
2
3
49. }}
47
50
53. }}
10.2
33
100
1
6
458-45 8-90 8 Triangles
547