Transcript Slide 1
Section 10.3
Goal
Find the side lengths of 30 ˚ -60 ˚ -90 ˚ triangles.
Key Vocabulary
30 ˚ -60 ˚ -90 ˚ triangles
Investigation
The second special right triangle is the 30 ˚ -60 ˚ -90 ˚ right triangle, which is half of an equilateral triangle.
Let’s start by using a little deductive reasoning to reveal a useful relationship in 30
˚
-60
˚
-90
˚
right triangles.
Investigation
Triangle
ABC
is equilateral, and segment
CD
is an altitude.
1.
2.
What are
m
∠
A
and
m
∠
B
?
What are
m
∠
ADC m
∠
BDC
?
and 3.
4.
What are
m
∠
ACD m
∠
BCD
?
and Is Δ
ADC
≅ Δ
BDC
? Why?
5.
Is
AD
=
BD
? Why?
Investigation
Notice that altitude
CD
divides the equilateral triangle into two right triangles with acute angles that measure 30 ° and 60°. Look at just one of the 30 ˚ 60 ˚ -90 ˚ right triangles. How do
AC
and
AD
compare?
Conjecture: In a 30 °-60°-90° right triangle, if the side opposite the 30 ° angle has length
x
, then the hypotenuse has length -?-.
2x
Investigation
Find the length of the indicated side in each right triangle by using the conjecture you just made.
18 33 17 8.5
10 26
Investigation
Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.
6 6 √ 3 8 4 √ 3 5 √ 3 8 50 √ 3 10 4 50
Investigation
You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.
In a 30 ˚ -60 ˚ -90 ˚ triangle: short side = x hypotenuse = 2x long side = x √ 3
Theorem 10.2 30
˚
-60
˚
-90
˚
Theorem Triangle
30 °-60°-90° Triangle Theorem
In a 30 °-60°-90° triangle, the length of the hypotenuse ℎ is 2 times the length of the shorter leg x, and the length of the longer leg is x √ 3 times the length of the shorter leg.
Example: hypotenuse longer leg shorter leg 3
30°-60°-90° Special Right Triangle
In a triangle
30°-60°-90°
, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.
Longer Leg
x
3
30 ° Hypotenuse
2x
60 °
x
Shorter Leg Example: 30 °
10 cm 5 cm
60 °
Find the value of a and b.
cm 7 cm 60 ° b b = 14 cm a 30 °
x
3
30 °
2x x
60 ° Step 1:
Find the missing angle measure.
30 ° Step 2:
Match the 30 °-60°-90° pattern with the problem
.
Step 3:
From the pattern, we know that
x = 7 , b = 2x, and a = x .
Step 4:
Solve for a and b
Example 1 Find Leg Length In the diagram,
PQR
is a 30° –60° –90° triangle with PQ = 2 and PR = 1 . Find the value of
b
.
SOLUTION You can use the Pythagorean Theorem to find the value of
b
.
(leg) 2 + (leg) 2 = (hypotenuse) 2
Write the Pythagorean Theorem.
1
2 +
b
2 =
2
2
Substitute.
1 +
b
2 = 4
b
2 = 3
Simplify.
Subtract 1 from each side.
b
= 3
Take the square root of each side.
Example 2 Find Hypotenuse Length In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the hypotenuse.
SOLUTION The hypotenuse of a
30° –60° –90°
triangle is twice as long as the shorter leg.
hypotenuse = 2
·
shorter leg = 2
·
12 = 24
30° –60° –90° Triangle Theorem Substitute.
Simplify.
ANSWER The length of the hypotenuse is 24 .
Example 3 Find Longer Leg Length In the 30° –60° –90° triangle at the right, the length of the shorter leg is given. Find the length of the longer leg.
SOLUTION The length of the longer leg of a
30° –60° –90°
triangle
longer leg = shorter leg
·
= 5
·
3 3
30° –60° –90° Triangle Theorem Substitute.
ANSWER The length of the longer leg is 5 3 .
Your Turn:
Find the value of
x
. Write your answer in radical form.
1.
ANSWER 14 2.
ANSWER 3 3 3.
ANSWER 10 3
Example 4 Find Shorter Leg Length In the 30° –60° –90° triangle at the right, the length of the longer leg is given. Find the length
x
of the shorter leg. Round your answer to the nearest tenth.
SOLUTION The length of the longer leg of a
30
°
–60
is the length of the shorter leg times .
°
–90
° triangle
longer leg = shorter leg
·
3
30° –60° – 90° Triangle Theorem
5 3 5 = x · =
x
2.9 ≈
x
ANSWER
3
Substitute.
Use a calculator.
The length of the shorter leg is about 2.9
.
Example 5 Find Leg Lengths In the 30° –60° –90° triangle at the right, the length of the hypotenuse is given. Find the length
x
of the shorter leg and the length
y
of the longer leg.
SOLUTION Use the
30
°
–60
°
–90
° Triangle Theorem to find the length of the shorter leg. Then use that value to find the length of the longer leg.
Shorter leg
hypotenuse = 2
·
shorter leg 8 = 2 · x 4 =
x
Longer leg
longer leg = shorter leg
·
y
= 4
·
3
y
= 4 3 3
Example 5 Find Leg Lengths ANSWER The length of the shorter leg is x = 4 .
The length of the longer leg is y = 4 .
Your Turn:
Find the value of each variable. Round your answer to the nearest tenth.
1.
ANSWER 3.5
2.
ANSWER x = 21 ;
Your Turn:
Find BC.
A.
4 in.
B.
8 in.
C.
D.
12 in.
Assignment
Pg. 552 - 555 #1 – 49odd