8.2 Special Right Triangles Geometry Objectives/Assignment Find the side lengths of special right...
-
Upload
collin-nash -
Category
Documents
-
view
218 -
download
1
Transcript of 8.2 Special Right Triangles Geometry Objectives/Assignment Find the side lengths of special right...
8.2 Special Right Triangles
Geometry
Objectives/Assignment
• Find the side lengths of special right triangles.
• Use special right triangles to solve real-life problems
• Quiz Next Class Period over 8.1- 8.2
Side lengths of Special Right Triangles• Right triangles whose angle
measures are 45°-45°-90° or 30°-60°-90° are called special right triangles.
Theorem 8.6: 45°-45°-90° Triangle Theorem• In a 45°-45°-90° triangle, the
hypotenuse is √2 times as long as each leg.
x
x√2x
45°
45°
Hypotenuse = √2 ∙ leg
Theorem 8.7: 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.
√3x
60°
30°
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
2xx
Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
• Find the value of x• 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°
Hypotenuse = √2 ∙ leg
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.• 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
Hypotenuse = √2 ∙ leg
Ex. 2: Finding a leg in a 45°-45°-90° Triangle
5
x x
Statement:Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
Reasons:45°-45°-90° Triangle Theorem
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
Ex. 3: Finding side lengths in a 30°-60°-90° Triangle
• Find 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=
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
X = 7 Y = 7√3
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
Y= 5 x = 10
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
X =
Y = 8√3
X =6 √2
Y = √2*6 √2 Y = 6√4Y = 6*2Y = 12
The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.
Hypotenuse = √2 ∙ leg
X =4 √3
Y = 2*4 =8
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
8 =√2* LegLeg = 5.6
X =5.6 √3 = 9.8Y = 2*5.6 =11.4
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.
Hypotenuse = √2 ∙ leg
Diagonal = 10√2 Diagonal = 14.1 in
The triangle is a 45°-45°-90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg.
Hypotenuse = √2 ∙ leg