Post on 15-Dec-2015
Objectives/DFA/HWObjectives/DFA/HWObjectives SWBAT use properties of 45o-45o-90o &
30o-60o-90o triangles.Why? Ex. – To find the distance from
home plate to 2nd on a baseball diamond.
DFA – p.504 #18
HW – pp.503-505 (2-32 even, 34-37 all)
Side lengths of Special Right TrianglesRight triangles whose angle measures are
45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.
Theorem 8.5: 45°-45°-90° Triangle TheoremIn a 45°-45°-90°
triangle, the hypotenuse is √2 times as long as each leg.
x
x√2x
45°
45°
Hypotenuse = √2 ∙ leg
Ex. 1: Finding the hypotenuse in a 45°-45°-90° TriangleFind the value of xBy the Triangle Sum
Theorem, the measure of the third angle is 45°. The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.
3 3
x
45°
Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
Hypotenuse = √2 ∙ leg
x = √2 ∙ 3
x = 3√2
3 3
x
45°
45°-45°-90° Triangle Theorem
Substitute values
Simplify
Ex. 2: Finding a leg in a 45°-45°-90° Triangle
Find the value of x.Because the triangle
is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°-90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg.
5
x x
Ex. 2: Finding a leg in a 45°-45°-90° Triangle
Statement:Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
Reasons:45°-45°-90° Triangle Theorem
5
x x
5
√2
√2x
√2=
5
√2x=
5
√2x=
√2
√2
5√2
2x=
Substitute values
Divide each side by √2
Simplify
Multiply numerator and denominator by √2
Simplify
Theorem 8.6: 30°-60°-90° Triangle TheoremIn 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. √3x
60°
30°
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
2xx
Ex. 3: Finding side lengths in a 30°-60°-90° TriangleFind the values of s
and t.Because the
triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
5
st
30°
60°
Ex. 3: Side lengths in a 30°-60°-90° Triangle
Statement:Longer leg = √3 ∙ shorter
leg
5 = √3 ∙ s
Reasons:30°-60°-90° Triangle Theorem
5
√3
√3s
√3=
5
√3s=
5
√3s=
√3
√3
5√3
3s=
Substitute values
Divide each side by √3
Simplify
Multiply numerator and denominator by √3
Simplify
5
st
30°
60°
The length t of the hypotenuse is twice the length s of the shorter leg.
Statement:Hypotenuse = 2 ∙ shorter
leg
Reasons:30°-60°-90° Triangle Theorem
t 2 ∙ 5√3
3= Substitute values
Simplify
5
st
30°
60°
t 10√3
3=
Using Special Right Triangles in Real Life
Example 4: Finding the height of a ramp.Tipping platform. A tipping platform is a
ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?
Solution:When the angle of elevation is 30°, the
height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet.
80 = 2h 30°-60°-90° Triangle Theorem40 = h Divide each side by 2.
When the angle of elevation is 30°, the ramp height is about 40 feet.
Solution:When the angle of elevation is 45°, the
height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet.
80 = √2 ∙ h 45°-45°-90° Triangle Theorem
80
√2= h Divide each side by √2
Use a calculator to approximate56.6 ≈ hWhen the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.