8.2-1 Special
Right Triangles
Geometry
Mr. Peebles
Spring 2013
Bell Ringer
• Determine if the three sides given to a
triangle form a triangle that’s Acute, Right,
or Obtuse? 6, 7, and 9.
Bell Ringer
• Determine if the three sides given to a
triangle form a triangle that’s Acute, Right,
or Obtuse? 6, 7, and 9.
• Answer:
Acute
Assignment-Due Now
• Pgs. 420-423 (11-17 Odds, 21-29
All, 31, 54, 59, 60)
Daily Learning Target (DLT)
• “I can apply the properties of 45-45-90
right triangles in mathematical and realworld problems.”
Side lengths of Special
Right Triangles
• Right 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 9.8: 45°-45°-90°
Triangle Theorem
• In a 45°-45°-90°
triangle, the
hypotenuse is √2
times as long as
each leg.
45°
√2x
x
45°
x
Hypotenuse = √2 ∙ leg
Ex. 1: Finding the hypotenuse in a
45°-45°-90° Triangle
• Find the value of x
• By 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
45°
x
Ex. 1: Finding the hypotenuse in a
45°-45°-90° Triangle
3
3
45°
x
Hypotenuse = √2 ∙ leg
45°-45°-90° Triangle
Theorem
x = √2 ∙ 3
Substitute values
x = 3√2
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°
5
Triangle
x
Statement:
Reasons:
Hypotenuse = √2 ∙ leg
5 = √2 ∙ x
5
√2
5
√2
√2
√2
x
5
√2
5√2
2
=
√2x
√2
= x
45°-45°-90° Triangle Theorem
Substitute values
Divide each side by √2
Simplify
= x
Multiply numerator and
denominator by √2
= x
Simplify
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 45° angle?
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
80
√2
= h
45°-45°-90° Triangle Theorem
Divide each side by √2
56.6 ≈ h
Use a calculator to approximate
When the angle of elevation is 45°, the ramp
height is about 56 feet 7 inches.
Assignment:
• Pgs. 428-429 (1-8, 29)
Exit Quiz
• Determine if the three sides given to a
triangle form a triangle that’s Acute, Right,
or Obtuse? 7, 9, and 13.