Aim #20 What is the relationship between the sides in a 30-60-90 triangle?
CC Geo
Do Now: The two legs of a right triangle are 12 and 15. What is the length of the
hypotenuse?
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30 -60 -90 Triangle
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ABC is an equilateral triangle with side lengths 2. Each angle measures ___ ,
Draw altitude BD.
B
A
C
a. Are the two right triangles congruent?
Explain.
b. What is the length of the shorter leg of each of the right triangles?
Explain.
c. Use the Pythagorean theorem to determine the length of the altitude.
d) Repeat this process for the three equilateral triangles below, with sides of
B
10, 4 and 8.
B
B
8
4
10
A
C
C
A
C
A
What relationship between the sides do you notice in these examples?
Conclusion
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Complete the following ratios for one of the 30 -60 -90 triangles:
shorter leg: hypotenuse = ___ : ___
shorter leg: longer leg = ___ : ___
30
2x
x√3
60
x
Exercise (2):
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Each row shows the side lengths of a different 30 -60 -90 triangle.
Complete the table with the missing lengths in simplest radical form.
shorter leg
longer leg
hypotenuse
25
40
10.5
12√3
Examples: Find the lengths of the 2 missing sides in the given 30-60-90 triangles.
1)
2)
12
30
20
3)
30
30
7
4)
5)
60
6)
60
8 3
30
5 3
7)
42
9 3
8)
9)
30
6 3
60
60
6
10)
11)
12)
8
20
60
30
30
12
13)
14)
15)
60 o
600
30 o
16) What is the exact area of an equilateral triangle with a side of 12?
17) What is the exact area of an equilateral triangle with a side of 20?
Let's Sum It Up!
In a 30-60-90 triangle, the hypotenuse is always twice the length
of the side opposite the 30 degree angle, and the side opposite the
60 degree angle is √3 times the side opposite the 30 degree angle.