5.8 Special Triangles WS

Transcription

Name
Date
Class
Practice A
LESSON
5-8
Applying Special Right Triangles
"
1. The sum of the angle measures in a triangle is 180°.
Find the missing angle measure. Then use the
Pythagorean Theorem to find the length

of the hypotenuse.
!
X
45°; 2
#
In a 45°-45°-90° triangle, the legs have equal length and the hypotenuse
is the length of one of the legs multiplied by 2 . Find the value of x.
2.
3.
2
X
45°
2
45°
45° X
4.
X
10
45°
45°
10qi
2
4qi
2
2 2
45°
X
4
10

5. Find the missing angle measure. Then use the
Pythagorean Theorem to find the length of the
hypotenuse.
60°; 3
In a 30°-60°-90° triangle, the hypotenuse is the length of the shorter
leg multiplied by 2, and the longer leg is the length of the shorter leg
multiplied by 3 . Find the values of x and y.
60°
8
7qi
3
30°
X
X
30°
60°
Y
6. x 4
y
4 3
7. x X
10
Y
60°
7
14
y
8. x 30°
Y
10 3
y
20
For Exercises 9 and 10, use a calculator to find each answer.
9. Andre is building a structure out of playing cards. Each card
is 6.3 centimeters long. He tries leaning the cards against
each other so that the angle at the top is 90°. Find the
distance between the edges of the cards to the nearest tenth.
10. Andre tries leaning the cards against each
other so the angle at the top is 60°. Find the
height x of the tops of the cards.
6.3 cm
90°
6.3 cm
90°
8.9 cm

CM

X
CM

CM
5.5 cm
11. Tell whether Andre can lay another card across the peaks of
the structures he built in Exercises 9 and 10. Possible answer:
Andre cannot lay a
card across the top of the structure in Exercise 9 because 6.3 cm 8.9 cm.
He can probably not lay a card across the top of the structure in
Exercise 10 because 6.3 cm is the distance between two consecutive
peaks, and there should be some overlap for the card to stay.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
59
Holt Geometry
Name
Date
Class
Name
Practice A
LESSON
5-8
5-8
�
1. The sum of the angle measures in a triangle is 180°.
Find the missing angle measure. Then use the
Pythagorean Theorem to find the length
��
of the hypotenuse.
�
�
�
45°; �2
�
3.
�
45°
2
45° �
4.
�
45°
10��
2
4��
2
2� 2
16
�
45°
60°; �3
30°
�
�
4
6. x �
y�
�
4� 3
7. x �
�
10
�
60°
60°
7
14
y�
8. x �
30°
�
�
10� 3
20
y�
9. Andre is building a structure out of playing cards. Each card
is 6.3 centimeters long. He tries leaning the cards against
each other so that the angle at the top is 90°. Find the
distance between the edges of the cards to the nearest tenth.
10. Andre tries leaning the cards against each
other so the angle at the top is 60°. Find the
height x of the tops of the cards.
90°
6.3 cm
90°
�
��
5.5 cm
triangle whose hypotenuse is the length of one of the legs of the larger
�
57�2 inches or about 10 inches, so
triangle. The height of the alcove is _____
8
He can probably not lay a card across the top of the structure in
Exercise 10 because 6.3 cm is the distance between two consecutive
the statues could have been placed in the alcoves.
peaks, and there should be some overlap for the card to stay.
Name
LESSON
5-8
Date
Holt Geometry
Class
Name
LESSON
5-8
Multiply and simplify. Assume a and b are nonnegative.
a�b
2
�
30°
�
4.
5.
60°
Theorem
�
��4
Example
45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, both legs are
congruent and the length of the hypotenuse
�
is � 2 times the length of a leg.
��4
�°
2�°
�
Holt Geometry
Class
2
Find the value of x in each figure. Give your answers in simplest radical form.
3.
Date
Reteach
a �b
�
2. (a � �b)(a � � b) �
60
Practice C
a � b)(��
a � b) �
1. (��
3
y�
perpendicular to the hypotenuse. This makes another smaller 45°-45°-90°
Andre cannot lay a
card across the top of the structure in Exercise 9 because 6.3 cm � 8.9 cm.
59
�
�3
6. x �
Possible answer: To find the height of a 45°-45°-90° triangle, draw a
11. Tell whether Andre can lay another card across the peaks of
the structures he built in Exercises 9 and 10. Possible answer:
�
8� 3
y�
inches
16
tall. She wonders whether the statues might have been placed in the alcoves. Tell
whether this is possible. Explain your answer.
��
��
��
�
4�3
8. Lucia also finds several statues around the building. The statues measure 9
8.9 cm
��
��
5. x �
Possible answer: Lucia’s hypothesis cannot be correct. The base of the
�
57�2 inches or just over 20 inches long, so a 22 _1_-inch tablet
alcove is _____
4
8
could not fit.
7
___
For Exercises 9 and 10, use a calculator to find each answer.
6.3 cm
�
20� 3
7. Around the perimeter of the building, Lucia finds small alcoves at regular intervals carved
into the stone. The alcoves are triangular in shape with a horizontal base and two sloped
equal-length sides that meet at a right angle. Each of the sloped sides measures 14 _1_
4
inches. Lucia has also found several stone tablets inscribed with characters. The stone
tablets measure 22 _1_ inches long. Lucia hypothesizes that the alcoves once held the stone
8
tablets. Tell whether Lucia’s hypothesis may be correct. Explain your answer.
7��
3
�
y�
2� °
2��
3
Lucia is an archaeologist trekking through the jungle of the Yucatan
Peninsula. She stumbles upon a stone structure covered with creeper
vines and ferns. She immediately begins taking measurements of her
discovery. (Hint: Drawing some figures may help.)
�
8
30
4. x �
�
�
�°
60°
�
In a 30°-60°-90° triangle, the hypotenuse is the length of the shorter
leg multiplied by 2, and the longer leg is the length of the shorter leg
�
multiplied by �3. Find the values of x and y.
30°
�
�
�
30°
��
�
2
12
10��
3
10
�
�
7� 2
____
2
Find the values of x and y. Give your answers in simplest radical form.
4
5. Find the missing angle measure. Then use the
Pythagorean Theorem to find the length of the
hypotenuse.
60°
�
�
�
45°
�
10
45°
45°
2��
2
7
�
45° �
In a 45°-45°-90° triangle, the legs have equal length and the hypotenuse
�
is the length of one of the legs multiplied by �2. Find the value of x.
2
Find the value of x in each figure. Give your answer in simplest
radical form.
2.
3.
1. 8��2
��
�
Class
Practice B
LESSON
2.
Date
��
��
�
�
��
��
��
��
��
�
��
��4
�
4
6.
�
2�3 � 2
7.
�
�
8�3 � 12
8.
30°
4�2 � 4
1
1
�
�2
��
_1_
�
Use the 45°-45°-90° Triangle Theorem to find the value of x in �EFG.
2
Every isosceles right triangle is a 45°-45°-90° triangle. Triangle
EFG is a 45°-45°-90° triangle with a hypotenuse of length 10.
Greg is a modeling enthusiast. He is working on modeling some
geometric shapes, but he finds he doesn’t have a ruler to take
measurements. In Greg’s desk drawer, he finds a protractor, a
straightedge, and a pencil. For Exercises 9 and 10, use 30°-60°-90°
and/or 45°-45°-90° triangles to accomplish each task.
�
�
�
Rationalize the denominator.
2.
��
�
��
��
��
�
��
�
�
x � 17 �2
��
��
�
� �
�
��
�
3.
x � 22 � 2
4.
�
�
�
�
30°-60°-90° triangle. The shorter leg of this second triangle then has
length _1_x. Use that leg as the longer leg of a third 30°-60°-90° triangle.
3
This smallest triangle has sides that are exactly one-third the length of
the original.
�
�
�
Divide both sides by �2.
1.
��
Possible answer: Name the length of the longer leg in �
�
�3 x.
��
a 30°-60°-90° triangle x. The shorter leg has length ___
3
Use the shorter leg of the original triangle as the longer leg of another
61
Hypotenuse is �2 times the length of a leg.
Find the value of x. Give your answers in simplest radical form.
Possible answer: Use one of the legs of the original 45°-45°-90° triangle
as the shorter leg of a 30°-60°-90° triangle. The hypotenuse of the
30°-60°-90° triangle will then have twice the length of one of the legs
of the 45°-45°-90° triangle. Then draw a 45°-45°-90° triangle with a leg
as the hypotenuse of the 30°-60°-90° triangle. This larger 45°-45°-90°
triangle has legs with exactly twice the length of the original 45°-45°-90°
triangle.
10. Describe how Greg can draw an exact 1 : 3 replica of a 30°-60°-90°
triangle. Sketch an example.
��
�
�
10 � x � 2
�
x� 2
10 � ____
___
�
�
�2
�2
�
5� 2 � x
9. Describe how Greg can draw an exact 2 : 1 replica of a 45°-45°-90° triangle.
That is, he will draw a triangle that has double the length of each side in the
original triangle. (Hint: Look back at Exercise 8.)
� ��
�
45°
60°
�
��
�
45°
�
��4
In a 45°-45°-90° triangle, if a leg
length is x, then the hypotenuse
�
length is x �2.
�
x � 4 �2
Holt Geometry
81
��
��
x � 25
62
Holt Geometry
Holt Geometry