1. Special Right Triangles
LESSON 8–3
Special Right Triangles

2. Five-Minute Check (over Lesson 8–2) TEKS 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
Lesson Menu

3. Find x.
A. 5 B. C.
D. 10.5
5-Minute Check 1

4. Find x. A. B.
C. 45
D. 51
5-Minute Check 2

5. 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
5-Minute Check 3

6. 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
5-Minute Check 4

7. 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
5-Minute Check 5

8. Which of the following are the lengths of an acute triangle?
B. 4, 7 , 8
C. 0.7, 2.4, 2.5
D. 36, 48, 62 __ 1 2
5-Minute Check 6

9. Mathematical Processes G.1(E), G.1(F)
Targeted TEKS
G.9(B) Apply the relationships in special right triangles 30º-60º-
90º and 45º-45º-90º and the Pythagorean theorem, including
Pythagorean triples, to solve problems.
Mathematical Processes
G.1(E), G.1(F)
TEKS

10. You used properties of isosceles and equilateral triangles.
Use the properties of 45°-45°-90° triangles.
Use the properties of 30°-60°-90° triangles.
Then/Now

11. Concept

12. 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.
Example 1

13. 45°-45°-90° Triangle Theorem
Find the Hypotenuse Length in a 45°-45°-90° Triangle
45°-45°-90° Triangle Theorem
Substitution
Example 1

14. 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.
Example 1

15. 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
Example 1

16. A. Find x.
A. 3.5
B. 7 C. D.
Example 1

17. B. Find x. A. B.
C. 16
D. 32
Example 1

18. 45°-45°-90° Triangle Theorem
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
Example 2

19. Rationalize the denominator.
Find the Leg Lengths in a 45°-45°-90° Triangle
Divide each side by
Rationalize the denominator.
Multiply.
Divide.
Example 2

20. Find b. A.
B. 3 C. D.
Example 2

21. Concept

22. 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.
Example 3

23. Find the length of the longer side.
Find Lengths in a 30°-60°-90° Triangle
Find the length of the longer side.
30°-60°-90° Triangle Theorem
Substitution
Simplify.
Example 3

24. Find the length of hypotenuse.
Find Lengths in a 30°-60°-90° Triangle
Find the length of hypotenuse.
30°-60°-90° Triangle Theorem
Substitution
Simplify.
Answer: x = 4,
Example 3

25. Find BC.
A. 4 in.
B. 8 in. C.
D. 12 in.
Example 3

26. 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?
Example 4

27. Use Properties of Special Right Triangles
Analyze 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.
Formulate Find the length of one side of the isosceles right triangle, and use the 45°-45°-90° Triangle Theorem to find the hypotenuse.
Example 4

28. So the side length is 1.5 inches.
Use Properties of Special Right Triangles
Determine 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
Example 4

29. Answer: The side length is 1.5 inches and the hypotenuse is
Use Properties of Special Right Triangles
Answer: The side length is 1.5 inches and the hypotenuse is
Justify Use the Pythagorean Theorem to check the dimensions of the triangle. ?
= 4.5 ?
4.5 = 4.5 
Evaluate We used the properties of special triangles. Our answer seems reasonable.
Example 4

30. 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.
Example 4

31. Special Right Triangles
LESSON 8–3
Special Right Triangles