#### Solved Examples and Worksheet for Special Right Triangles

Q1If the length of the leg opposite to the 60° angle in a 30° - 60° - 90° triangle is 93 cm, then the length of the hypotenuse is
A. 33 cm.
B. 9 cm.
C. 6 cm.
D. 18 cm.

Step: 1
If d is the length of the hypotenuse in a 30°- 60° - 90° triangle, then the length of the side opposite the 60° angle is d32 . Step: 2
So, d32 = 93
Step: 3
d = 9323 = 18 cm
Correct Answer is :   18 cm.
Q2If the length of the leg of a 45° - 45° - 90° triangle is 15 ft, then the length of the hypotenuse is
A. 92ft
B. 9 ft
C. 152 ft.
D. 3 ft

Step: 1
If S is the length of the leg of a 45° - 45° - 90° triangle, then the length of the hypotenuse is S2 = 152 ft. Correct Answer is :   152 ft.
Q3If the length of the hypotenuse of a 45°- 45° - 90° triangle is 10 cm then the length of each leg is

A. 8 cm.
B. 52 cm.
C. 43 cm.
D. 83 cm.

Step: 1
If S is the length of the leg of a 45° - 45° - 90° triangle, then the length of the hypotenuse is S2. Step: 2
So, S2 = 10
Step: 3
S = 102
Step: 4
S = 1022 ×2 = 1022 = 52 cm
Correct Answer is :   52 cm.
Q4What is the relation between the length of the shortest side and hypotenuse in a 30o-60o-90o triangle?

A. shortest side is √32th of the hypotenuse
B. shortest side is half of the hypotenuse
C. shortest side is 34th of the hypotenuse
D. shortest side is 14th of the hypotenuse

Step: 1
In a 30o - 60o - 90o triangle, the length of the shortest side is half of its hypotenuse.
Correct Answer is :   shortest side is half of the hypotenuse
Q5The length of PR in the figure is 72 inches. What are the lengths of PQ¯ and QR¯? A. 8 inches and 9 inches
B. 7 inches and 7 inches
C. 8 inches and 7 inches
D. 7 inches and 9 inches

Step: 1
The triangle is a 45o - 45o - 90o triangle.
Step: 2
Length of PR = 72 inches.
Step: 3
In a 45o - 45o - 90o triangle, length of hypotenuse is 2 times the length of leg.
Step: 4
In ΔPQR, PQ¯ and QR¯ are congruent legs and PR¯ is the hypotenuse.
Step: 5
PR = PQ2
Step: 6
PQ = PR2
[Divide each side by 2.]
Step: 7
PQ = 722
[Replace PR with 72.]
Step: 8
PQ = 7 inches
[Simplify.]
Step: 9
Since the lengths of two legs are equal in 45o - 45o - 90o triangle, PQ = QR = 7 inches.
Step: 10
The lengths of PQ¯ and QR¯ are 7 inches.
Correct Answer is :   7 inches and 7 inches
Q6The lengths of two legs are equal and hypotenuse is 2 times the length of a leg in a triangle. Which of the following are the angles of the triangle?

A. 60o - 60o - 60o
B. 30o - 60o - 90o
C. 45o - 45o - 90o
D. 90o - 90o - 90o

Step: 1
In a 45o - 45o - 90o triangle, the lengths of two legs are same and length of hypotenuse is 2 times the length of a leg.
Step: 2
The triangle is a 45o - 45o - 90o triangle.
Correct Answer is :   45o - 45o - 90o
Q7Is a triangle with side lengths 3, 6 and 33 a 30o-60o-90o triangle?

A. no, it is not a right triangle
B. yes
C. no, it is an equilateral triangle
D. no, it is a 45o- 45o- 90o triangle

Step: 1
The side lengths of the triangle are 3, 6 and 33.
Step: 2
62 = 32 + (33) 2
[Checking for Pythagorean theorem.]
Step: 3
Since given measures satisfy Pythagorean theorem, they form a right triangle.
Step: 4
Here, the length of hypotenuse is 6 and the length of shorter leg is 3.
Step: 5
Since the length of hypotenuse is twice the length of shorter leg, the sides form a 30o - 60o - 90o triangle.
Q8Find the missing sides in the triangle. [a = 5 cm.] A. BC = 10 cm and AC = 252 cm
B. BC = 52 cm and AC = 5 cm
C. BC = 5 cm and AC = 52 cm
D. BC = 4 cm and AC = 7 cm

Step: 1
Since two of the angles of the triangle are equal and one of the angle is 90o, the triangle is an isosceles right triangle.
Step: 2
In an isosceles right triangle,the legs have same length and hypotenuse is 2 times its leg.
Step: 3
AB¯, BC¯ are legs and AC¯ is the hypotenuse from the figure.
Step: 4
BC = AB = 5 cm
[Since legs have same lengths in an isosceles right triangle.]
Step: 5
AC = BC × 2 cm = 52 cm
[Substitute BC = 5.]
Step: 6
The missing sides are BC = 5 cm and AC = 52 cm.
Correct Answer is :   BC = 5 cm and AC = 52 cm
Q9What is the length of side RP¯ in ΔPQR? [a = 16.] A. 18 ft
B. 16 ft
C. 17 ft
D. 20 ft

Q10The area of a 30o - 60o- 90o triangle is 73 units2. Find the length of the shorter leg.

A. 3 units
B. 3.74 units
C. 7.48 units
D. 3.74 3 units

Step: 1
Let x units be the length of the shorter side. Step: 2
[30o-60o-90o-triangle theorem.]
Step: 3
Area of the triangle = (12) × base × height
[Formula.]
Step: 4
73 = 12 × x × x3
[Substitute.]
Step: 5
x2 = 14
[Simplify.]
Step: 6
x = 3.74
[Take square root on both sides.]
Step: 7
Length of the shorter leg is 3.74 units.
Correct Answer is :   3.74 units
Q11The area of 45o- 45o - 90o triangle is 8 in.2. What is the length of the hypotenuse?

A. 43 in.
B. 4 in.
C. 8 in.
D. 42 in.

Step: 1
Let x can be the length of the leg Step: 2
[45o-45o-90o Triangle Theorem.]
Step: 3
Area of the triangle = 12 × base × height
[Formula.]
Step: 4
8 = 12 × x × x
[Substitute.]
Step: 5
x2 = 16
[Simplify.]
Step: 6
x = 4
[Take square roots on Both sides.]
Step: 7
Hypotenuse = x2 = 42 in.
[Step2 and step6.]
Correct Answer is :   42 in.
Q12The hypotenuse of a 30o - 60o - 90o triangle is 43 cm more than the shorter leg. What is the length of the longer leg?
A. 43 cm
B. 4 cm
C. 8 3 cm
D. 12 cm

Step: 1
Let x be the length of the shorter leg. Step: 2
[30o-60o-90o-triangle theorem.]
Step: 3
Hypotenuse = shorter leg + 43
[Given.]
Step: 4
2x = x + 43
[Substitute.]
Step: 5
x = 43
[Simplify.]
Step: 6
Length of the longer leg = x3 = 3 × 43 = 12 cm
[Step2 and step5.]
Correct Answer is :   12 cm