Download 8-3 Special Right Triangles

Five-Minute Check (over Lesson 8–2)
CCSS
Then/Now
Theorem 8.8: 45°-45°-90° Triangle Theorem
Example 1: Find the Hypotenuse Length in a 45°-45°-90°
Triangle
Example 2: Find the Leg Lengths in a 45°-45°-90° Triangle
Theorem 8.9: 30°-60°-90° Triangle Theorem
Example 3: Find Lengths in a 30°-60°-90° Triangle
Example 4: Real-World Example: Use Properties of Special
Right Triangles
Over Lesson 8–2
Find x.
A. 5
B.
C.
D. 10.5
Over Lesson 8–2
Find x.
A.
B.
C. 45
D. 51
Over Lesson 8–2
Determine whether ΔQRS with vertices
Q(2, –3), R(0, –1), and S(4, –1) is a right triangle.
If so, identify the right angle.
A. yes; S
B. yes; Q
C. yes; R
D. no
Over Lesson 8–2
Determine whether the set of numbers can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, obtuse, or right.
16, 30, 33
A. yes, acute
B. yes, obtuse
C. yes, right
D. no
Over Lesson 8–2
Determine whether the set of numbers can be the
measures of the sides of a triangle. If so, classify
the triangle as acute, obtuse, or right.
A. yes, acute
B. yes, obtuse
C. yes, right
D. no
Over Lesson 8–2
Which of the following are the lengths of an acute
triangle?
A. 25, 20, 15
1,8
B. 4, 7 __
2
C. 0.7, 2.4, 2.5
D. 36, 48, 62
Content Standards
G.SRT.6 Understand that by similarity, side
ratios in right triangles are properties of the
angles in the triangle, leading to definitions of
trigonometric ratios for acute angles.
Mathematical Practices
1 Make sense of problems and persevere in
solving them.
7 Look for and make use of structure.
You used properties of isosceles and
equilateral triangles.
• Use the properties of 45°-45°-90° triangles.
• Use the properties of 30°-60°-90° triangles.
Find the Hypotenuse Length in a 45°-45°-90°
Triangle
A. Find x.
The given angles of this triangle are 45° and 90°. This
makes the third angle 45°, since 180 – 45 – 90 = 45.
Thus, the triangle is a 45°-45°-90° triangle.
Find the Hypotenuse Length in a 45°-45°-90°
Triangle
45°-45°-90° Triangle Theorem
Substitution
Find the Hypotenuse Length in a 45°-45°-90°
Triangle
B. Find x.
The legs of this right triangle have the same measure,
x, so it is a 45°-45°-90° triangle. Use the
45°-45°-90° Triangle Theorem.
Find the Hypotenuse Length in a 45°-45°-90°
Triangle
45°-45°-90° Triangle Theorem
Substitution
x = 12
Answer: x = 12
A. Find x.
A. 3.5
B. 7
C.
D.
B. Find x.
A.
B.
C. 16
D. 32
Find the Leg Lengths in a 45°-45°-90° Triangle
Find a.
The length of the hypotenuse of a 45°-45°-90° triangle
is
times as long as a leg of the triangle.
45°-45°-90° Triangle Theorem
Substitution
Find the Leg Lengths in a 45°-45°-90° Triangle
Divide each side by
Rationalize the denominator.
Multiply.
Divide.
Find b.
A.
B. 3
C.
D.
Find Lengths in a 30°-60°-90° Triangle
Find x and y.
The acute angles of a right triangle are
complementary, so the measure of the third angle is
90 – 30 or 60. This is a 30°-60°-90° triangle.
Find Lengths in a 30°-60°-90° Triangle
Find the length of the longer side.
30°-60°-90° Triangle
Theorem
Substitution
Simplify.
Find Lengths in a 30°-60°-90° Triangle
Find the length of hypotenuse.
30°-60°-90° Triangle
Theorem
Substitution
Simplify.
Answer: x = 4,
Find BC.
A. 4 in.
B. 8 in.
C.
D. 12 in.
Use Properties of Special Right
Triangles
QUILTING A quilt has the design shown in the
figure, in which a square is divided into 8 isosceles
right triangles. If the length of one side of the square
is 3 inches, what are the dimensions of each
triangle?
Use Properties of Special Right
Triangles
Understand
You know that the length of the side of
the square equals 3 inches. You need
to find the length of the side and
hypotenuse of one isosceles right
triangle.
Plan
Find the length of one side of the
isosceles right triangle, and use the
45°-45°-90° Triangle Theorem to find
the hypotenuse.
Use Properties of Special Right
Triangles
Solve
Divide the length of the side of the
square by 2 to find the length of the
side of one triangle. 3 ÷ 2 = 1.5
So the side length is 1.5 inches.
45°-45°-90° Triangle
Theorem
Substitution
Use Properties of Special Right
Triangles
Answer:
The side length is 1.5 inches and the
hypotenuse is
Check
Use the Pythagorean Theorem to check
the dimensions of the triangle.
?
?
2.25 + 2.25 = 4.5
4.5 = 4.5 
BOOKENDS Shaina designed 2 identical bookends
according to the diagram below. Use special
triangles to find the height of the bookends.
A.
B. 10
C. 5
D.