Download Special Right Triangles

MATH II
Stephanie Ballantine & Altenese Gibbs
October 30, 2009
1st Block (White)
Special Right Triangles
1st Performance Standard MM2G1:
“Identify and use special right triangles.”
MM2G1a – Determine the lengths of sides of 30°-60°-90° triangles.
MM2G1b – Determine the lengths of sides of 45°-45°-90° triangles.
Special Right Triangles
{.Vocabulary.}
Theorem 5.1 45°-45°-90° Triangle Theorem:
In a 45°-45°-90° triangle, the hypotenuse is √2
times as long as each leg.
Theorem 5.2 30°-60°-90° Triangle Theorem:
In a 30°-60°-90° triangle, the hypotenuse is twice
as long as the shorter leg, and the longer leg is √3 times
as long as the shorter leg.
Find the value of x.
11
x
45°
x
x
2√6
Solution:
Solution:
a. Because the sum of the angle
b. You know that each of the two
measures in a triangle is 180°, the
congruent angles in the triangle
measure of the third angle is 45°.
has a measure of 45° because the
So, the triangle is a 45°-45°-90°
sum of the angle measures in a
triangle, and by the theorem 5.1,
triangle is 180°. So, the triangle is
the hypotenuse is √2 times as long
a 45°-45°-90° triangle.
as each leg.
Hypotenuse = leg x √2
x = 11√2
Hypotenuse = leg x √2
2√6 = X x √2
2√3 = X
Find the value of x and y.
Solution:
60°
x
y
STEP 1 Find the value of x.
longer leg = shorter leg x √3
4√3 = x√3
4=x
STEP 2 Find the value of y.
hypotenuse = 2 x shorter leg
y=2x4
y=8
30°
4√3
Examples:
x
x
12√2
• What is the length of the
legs? X = ?
• Use the formula.
(leg)
(hyp / √2)
X =
12√2
√2
X = 12 (The legs = 12)
6
6
x
• What is the length of the
hypotenuse? X = ?
• Use the formula.
(hypotenuse) (leg x √2)
X =
6 x √2
X = 6√2
(The hypotenuse = 6√2 ).
Special Right Triangles
2nd Performance Standard MM2G1:
“Define and apply sine, cosine, & tangent ratios of right triangles.”
MM2G2a – Discover the relationship of the trigonometric ratios for
similar triangles.
MM2G2b – Explain the relationship between the trig ratios of
complementary angles.
MM2G2c – Solve application problems using the trig ratios.
Apply the Tangent Ratio
{.Vocabulary.}
Trigonometry:
A branch of mathematics that
deals with the relationships
between the sides and angles of
triangles and the calculations
based on these relationships.
Trigonometric Ratio:
A ratio of the lengths of two
sides in a right triangle.
Tangent:
The ratio of the length of the
leg opposite an angle to the
length of the leg adjacent to the
angle that is constant for a given
angle measures.
Complementary Angles:
Two angles are like this if the
sum of their measures is 90°.
Sine:
The ratio of the length of the
leg opposite an angle to the
length of the hypotenuse to that
same angle.
Cosine:
The ratio of the length of the
leg adjacent an angle to the
length of the hypotenuse of that
same angle.
Solve a Right Triangle:
Find the measures of all its
sides and angles.
Find sin X and sin Y.
“hypotenuse is
always across
from the right
angle.”
Y
20
Z
52
48
sin X = opposite of angle X = YZ
hypotenuse
XY
Plug it into a calculator.
= 20 ~ 0.3846
52
sin Y = opposite of angle Y = XZ
hypotenuse
XY
Plug it into a calculator.
= 48 ~ 0.9231
52
X
Find cos X and cos Y.
“Adjacent is the
side next to the
angle you’re
solving for, unless
it’s the
hypotenuse.”
Y
20
Z
52
X
48
cos Y = adjacent of angle Y = YZ
hypotenuse
XY
Plug it into a calculator.
= 20 ~ 0.9231
52
cos X = adjacent of angle X = XZ
hypotenuse
XY
Plug it into a calculator.
= 48 ~ 0.3846
52
Find the sine and cosine of angles X, Y, L,
and M of the similar triangles. Then
M
compare the ratios.
Y
c
b
Z
2c
2b
a
X
N
2a
sin X = b
c
cos X = a
c
sin L = 2b
2c
cos L = 2a
2c
sin Y = a
c
cos Y = b
c
sin M = 2a
2c
cos M = 2b
2c
So, since triangle XYZ and LMN are similar triangles, sin X = sin L,
cos X = cos L, sin Y = sin M, and cos Y = cos M.
L
Find tan X and tan Y.
Y
13
Z
85
“Note that in the right triangle,
XYZ, angle X and Y are
complementary angles.”
X
84
tan X = opposite of angle X = YZ
adjacent to angle X
XZ
Plug it into a calculator.
= 13 ~ 0.1548
84
tan Y = opposite of angle Y = XZ
adjacent
YZ
Plug it into a calculator.
= 84 ~ 6.4615
13
Find tan X and Y for the similar triangles.
Then compare tangent ratios.
Y
Y
c
b
Z
2b
a
X
Z
tan X = b
c
tan Y = a
b
2c
2a
tan X = 2b = b
2a a
tan Y = 2a = a
2b b
The values of tan X and tan Y for the similar triangles are equivalent.
X
Example:
x
15
tan 28° = opposite
adjacent
tan 28° = X
15
15 x tan 28° = X
15(0.5317) ~ X
8.0 ~ X
28°
• What is the value of X?
• Use the tangent of an acute
angle to find a leg length.
• Write the ratio for tangent of
28°.
• Substitute.
• Multiply each side by 15.
• Use a calculator to find tan
28°.
• Simplify.
Solving Application
Problems
“Compare real life situations to the use of special right
triangles.”
The Eiffel Tower:
In this image we are given a picture
of the Eiffel tower. We are also
given the height(longer leg) of the
tower along with the width(shorter
leg). Now we are to find the length
across the bottom-right corner to
the top-left corner(hypotenuse).
1. Use the formula to find the hypotenuse.
Hyp = 2 x shorter leg
2 x 382 ft. = 764 ft.
Drive-in Movie: You are at a drivein movie with your friend. The
screen is level with the top of the
car. There is 68 ft. between the top
of your car and the top of the
screen. The angle of elevation from
the same distance is 43°.
1. What kind of triangle is this?
45°-45°-90° Triangle
2. Use the formula to find the
leg length.
Leg = hyp / √2
68 ft. / √2 ~ 48.08
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