Download 8.1 Similar Right Triangles

8.1 Similar Right Triangles
Objectives:
G.SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
For the Board: You will be able to solve problems involving similar right triangles formed by the
altitude drawn to the hypotenuse of a right triangle.
Anticipatory Set:
The geometric mean of two positive numbers a and b is the positive number x such that
Open the book to page 535 and read example 2.
Example: Find the geometric mean of each pair of numbers.
Write the answer in simplest radical form.
a. 4 and 25
b. 5 and 30
4/x = x/25
5/x = x/30
2
x = 100
x2 = 150
x = 10
x=5 6
Find the geometric mean of each pair of numbers.
Write the answer rounded to the nearest tenth.
c. 3.6 and 11.4
d. 7.1 and 18.7
3.6/x = x/11.4
7.1/x = x/18.7
x2 = 41.04
x2 = 132.77
x = 6.4
x = 11.5
White Board Activity:
Practice: Find the geometric mean of each pair of numbers.
Write the answer in simplest radical form.
a. 6 and 16
b. 10 and 30
6/x = x/16
10/x = x/30
2
x = 96
x2 = 300
x=4 6
x = 10 3
Find the geometric mean of each pair of numbers.
Write the answer rounded to the nearest tenth.
c. 4.2 and 8.1
d. 9.5 and 11.8
4.2/x = x/8.1
9.5/x = x/11.8
2
x = 34.02
x2 = 112.1
x = 5.8
x = 10.6
a x
 .
x b
C
Begin with a right triangle, drawn with its base the hypotenuse.
Label the short leg a, the long leg b, and the hypotenuse c.
Hint: make the left leg shorter than the right leg.
a
Draw an altitude from the right angle to the hypotenuse.
Hint: an altitude is drawn from a vertex perpendicular to the
opposite side.
Another name for altitude is height, label the altitude h.
This altitude divides the hypotenuse into two pieces.
Label the left piece x and the right piece y.
3 right triangles are formed in this picture:
a small, a medium, and a large.
x
B
b
h
y
D
A
c
Activity:
Large triangle: white
Small triangle: pink
Medium triangle: blue
1. Cut out the triangles and check to make sure the small and medium triangle will fit perfectly over
the large triangle.
2. Label the large triangle. Call the short leg a, the long leg b, and the hypotenuse c.
3. Transfer these values to the small and medium triangle where appropriate.
4. Draw the altitude on the large triangle, label it h.
5. Transfer this value to the small and medium triangle where appropriate.
6. On the large triangle, the altitude divides the hypotenuse into two pieces. Label the left piece x and
the right piece y.
7. Transfer these values to the small and medium triangle where appropriate.
Check your labels.
a
h
b
h
a
b
x
y
c
8. Since the small triangle and the large triangle are right triangles and share an angle, they are similar.
Arrange the small triangle and the large triangle side by side so that you can see which side of the
small triangle corresponds to which side of the large triangle.
Complete the ratio.
x
a

h
b

a
c
a
h
c
x
a
b
9. Since the medium triangle and the large triangle are right triangles and share an angle, they are similar.
Arrange the medium triangle and the large triangle side by side so that you can see which side of the
medium triangle corresponds to which side of the large triangle.
h
y
b
Complete the ratio.


a
b
c
a
c
b
h
b
y
10. The small triangle and the medium triangle will also be similar.
Arrange the small triangle and the medium triangle side by side so that you can see which side of the
small triangle corresponds to which side of the medium triangle.
x
h
a
Complete the ratio.


h
y
b
a
h
y
b
x
11. Each of these ratios contains a built-in geometric mean. Pull them out.
x
h
x
a
y



h
y
a
c
b
h
b
c
Now that you have seen that these proportions come from similar triangle relationships, let’s look for an
easier way of remembering them. We will look for patterns on the original picture.
Vocabulary
left
leg
left
piece
(a)
(b)
(h) altitude
(y)
(x)
Right
leg
right
piece
D
(c) hypotenuse
e
The altitude is the geometric mean between the two “pieces” of the hypotenuse.
left piece
altitude

altitude right piece
The leg is the geometric mean between its “piece” and the hypotenuse.
left piece
left leg
rightt piece
right leg


left leg
hypotenuse
right leg
hypotenuse
C
Example: Given right triangle ABC with altitude to the hypotenuse CD,
find each of the following values. Round answers to the
nearest tenth when appropriate.
b
A
a. x = 5, y = 8, find c, h, a, b.
c = 13
5/h = h/8
5/a = a/13
8/b = b/13
2
2
h = 40
a = 65
b2 = 104
h = 6.3
a = 8.1
b = 10.2
b. y = 14, c = 18, find y, h, a, b .
x=4
4/h = h/14
4/a = a/18
14/b = b/18
h2 = 56
a2 = 72
b2 = 252
h = 7.5
a = 8.5
b = 15.9
c. x = 6, c = 15, find x, h, a, b.
y=9
6/h = h/9
6/a = a/15
9/b = b/15
2
2
h = 54
a = 90
b2 = 135
h = 7.3
a = 9.5
b = 11.6
d. a = 8, c = 20, find b, x, y, h.
x/8 = 8/20
y = 16.8
16.8/b = b/20
3.2/h = h/16.8
2
20x = 64
b = 336
h2 = 53.76
x = 3.2
b = 18.3
h = 7.3
e. b = 10, c = 15, find a, x, y, h.
a2 + b2 = c2
a2 + 102 = 152
a2 + 100 = 225
a2 = 125
x/b = b/c
x/10 = 10/15
15x = 100
x = 6.7
y/a = a/c
y/11.2 = 11.2/15
15y = 125
y = 8.3
2
x/h = h/y
6.7/h = h/8.3
h = 55.61
h = 7.5
White Board Activity:
Practice:
a. x = 7, y = 10, find c, h, a, b.
c = 17
7/h = h/10
h2 = 70
h = 8.4
b. y = 18, c = 24, find y, h, a, b .
x=6
6/h = h/18
h2 = 108
h = 10.4
c. x = 4, c = 16, find x, h, a, b.
y = 12
4/h = h/12
h2 = 48
h = 6.9
d. a = 10, c = 24, find b, x, y, h.
x/10 = 10/24 y = 19.8
24x = 100
x = 4.2
7/a = a/17
a2 = 119
a = 10.9
10/b = b/17
b2 = 170
b = 13.0
6/a = a/24
a2 = 144
a = 10
18/b = b/24
b2 = 432
b = 20.8
4/a = a/16
a2 = 64
a=8
12/b = b/16
b2 = 192
b = 13.9
19.8/b = b/24
b2 = 475.2
b = 21.8
4.2/h = h/19.8
h2 = 83.16
h = 9.1
a
h
y
x
D
c
a = 11.2
B
e. b = 12, c = 18, find a, x, y, h.
a2 + b2 = c2
a2 + 122 = 182
x/b = b/c
x/12 = 12/18
y/a = a/c
y/13.4 = 13.4/18
x/h = h/y
8/h = h/13.4
a2 + 144 = 324
18x = 144
18y = 179.56
h2 = 107.2
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 537 – 539 prob. 6 – 14, 18 – 40, 42.
For a Grade:
Text: pgs. 537 – 539 prob. 8, 12, 18, 24, 36.
a2 = 180
x=8
y = 13.4
h = 10.4
a = 13.4