Unit 8 Lesson 9.4ASpecial Right Triangles
CCSSG-SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.G-SRT 8.1: Derive and use the trigonometric ratios for special right triangles (30°,60°,90°and 45°,45°,90°).
Lesson Goals• Define a special right triangle• Identify the patterns in special
right triangles.• Use the special right triangle
theorems to find missing side lengths in right triangles.
ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
Why are special right triangle theorems useful if we have the Pythagorean Theorem?
DefinitionSpecial Right Triangles
A right triangle with acute angles of 45o and 45o.
A right triangle with acute angles of 30o and 60o.
A
BC
45o
45o
D
FE
60o
30o
not in notes
You Try (not in notes)Find the missing angle measures.
7
7
xo
90x x
2 90x
45x
xo
You TryFind the missing side length. Leave answer in radical form.
2
2 c
2 2 22 2c 2 8c
8c
2 2c 45o
45o
You Try
7
7 c
Find c. Leave answer in radical form.
45o
45o
2
2
7
72 2 7 2
45o
45o
45o
45o
10
10 2
45o
45o
58 2
45o
45o
the hypotenuse is times as long as each leg.2
45o - 45o - 90o Triangle Theorem
2ss
s
A
BC
45o
45o
In a 45 -45 -90 triangle,
You Try
260o
30o
b 4
2 2 24 2 b 216 4 b
212 b
12b
2 3b
Find the missing side length. Leave answer in radical form.
You Try
760o
30o
7 3 c
22 27 7 3c 2 196c
196c
14c
Find the missing side length. Leave answer in radical form.
You Try
a60o
30o
21 3 42
22 242 21 3a
21764 1323a 2441 a
441a
21a
Find the missing side length. Leave answer in radical form.
260o
30o
2 3 4
760o
30o
7 3
14
2160o
30o
21 3
42
360o
30o
3 3 6
60o
30o
4 3 8
4
2n
the length of the hypotenuse is 2 the shorter leg
3n
D
FE
60o
30o
n
30o - 60o - 90o Triangle Theorem
In a 30°-60°-90° triangle,
and the longer leg is the shorter leg.3
Steps to help in solvingStep 1: Place the pattern on the triangle (make sure you use a different variable than the ones on the triangle)
Step 2: Setup 3 different equations based on what you have on each side of the triangle
Step 3: Solve the equation that only has one variable
Step 4: Use the answer from step 3 to solve other two sides.
Please note not all answers will be “pretty”
ExampleFind the value of x.
ExampleFind the value of x.
x
5 5
45o
5 2x
Example
12 x
45o
12 2x
12
2x
6 2 x
Find the value of x.
Example60o
30o
f
g
4 3g 4 2 4f
8
Find the value of f and g.
10 3
3
Example
5 3m
5
3m
5 3
3m
5 32
3
60o
30o
n
5
m
2n m
Find the value of m and n.
Summary
Why are special right triangle theorems useful if we have the Pythagorean Theorem?
Today’s Assignment
p. 554: 1 – 14, 43, 45, 47 – 49