# Right Triangle Worksheet

Right Triangle Worksheet
• Page 1
1.
State which of the following measures will form a right triangle.
I. 9, 12, 15
II. 12, 16, 20
III. 12, 15, 20
 a. I and II b. only III c. only II d. only I

#### Solution:

According to Pythagorean theorem, in a right triangle, square of the longest side = sum of the squares of the other two sides.

In I, 152 = 92 + 122

225 = 81 + 144 = 225

The measures, 9, 12, and 15 will form a right triangle.

In II, 202 = 122 + 162

400 = 144 + 256 = 400

The measures, 12, 16, and 20 will form a right triangle.

In III, 202 = 122 + 152

400 ≠ 144 + 225 = 369, 400 is not equal to 369

The measures, 12, 15, and 20 will not form a right triangle.

So, I and II only will form right triangles.

2.
State which of the following measures will form a right triangle.
(i) 6, 8, 10
(ii) 3, 4, 5
(iii) 2, 3, 5
 a. Both (i) and (ii) b. only (ii) c. only (i) d. only (iii)

#### Solution:

According to Pythagorean theorem, in a right triangle, square of the longest side = sum of the squares of the other two sides.

In (i), 102 = 62 + 82

100 = 36 + 64 = 100

The measures, 6, 8, and 10 will form a right triangle.

In (ii), 52 = 32 + 42

25 = 9 + 16 = 25

The measures, 3, 4, and 5 will form a right triangle.

Applying Pythagorean theorem for (iii), 52 ≠ 22 + 32

25 ≠ 4 + 9 = 13, 25 is not equal to 13

The measures 2, 3, and 5 cannot form a right triangle.

So, (i) and (ii) only will form right triangles.

3.
What are the angle measures of an isosceles right triangle?
 a. 30o, 60o and 90o b. 30o, 30o and 90o c. 45o, 45oand 90o d. 40o, 40o and 100o

#### Solution:

If two angles of a right triangle are same then the triangle is said to be an isosceles right triangle.

Sum of angles in a triangle = 180o

In right triangle one angle is 90o.

Sum of the other two angles is 90o.

Since other two angles are equal, each angle = 45o.
[Since 90o/2 = 45o.]

The angle measures in a right isosceles triangle are 45o, 45o and 90o.

4.
Is every isosceles triangle an isosceles right triangle?
 a. Yes b. No

#### Solution:

If the lengths of two sides are equal then it is an isosceles triangle.

In right triangle, if lengths of two sides are equal then it is an isosceles right triangle.

Every isosceles triangle is not an isosceles right triangle; for example, the lengths of sides 2, 2 and 3 is an isosceles triangle.

Since 32 ≠ 22 + 22, it is not a right triangle.

So, every isosceles triangle is not an isosceles right triangle.

5.
What type of triangle will be formed from two sides of a square field and diagonal walkway?

 a. 30o-60o-90o triangle b. 45o-45o-90o triangle c. 60o-60o-60o triangle d. None of the above

#### Solution:

Diagonal of the square bisects the angle.

The figure formed from the two sides of a square field and diagonal walkway is a 45o-45o-90o triangle.

6.
What is the relation between the length of the shortest side and hypotenuse in a 30o-60o-90o triangle?
 a. Shortest side is half of the hypotenuse b. Shortest side is three times of the hypotenuse c. Shortest side is double the hypotenuse d. Shortest side is $\sqrt{3}$ times of the hypotenuse

#### Solution:

The length of the shortest side is half of the hypotenuse in a 30o-60o-90o triangle.

7.
The length of PR in the figure is 7√2 inches. What are the lengths of PQ and QR?

 a. 4 inches and 4 inches b. 7 inches and 7 inches c. 2 inches and 3 inches d. 6 inches and 4 inches

#### Solution:

The triangle is a 45o-45o-90o triangle.

Length of PR = 7√2 inches.

In a 45o- 45o-90o triangle, length of hypotenuse is √2 times the length of leg.

In ΔPQR, PQ and QR are congruent legs and PR is the hypotenuse.

PR = PQ√2

PQ = PR/√2
[Divide each side by √2.]

PQ = 7√2/√2
[Replace PR with 7√2.]

PQ = 7 inches
[Simplify.]

Since the lengths of two legs are equal in 45o-45o-90o triangle, PQ = QR = 7 inches.

The lengths of PQ and QR are 7 inches and 7 inches.

8.
Is a triangle with side lengths 4, 8 and 4√3 a 30o-60o-90o triangle?
 a. No b. Yes

#### Solution:

The side lengths of the triangle are 4, 8 and 4√3.

82 = 42 + (4√3) 2
[Check for Pythagorean theorem.]

Since given measures satisfy Pythagorean theorem, they form a right triangle.

Here, the length of hypotenuse is 8 and the length of shorter leg is 4.

Since the length of hypotenuse is twice the length of shorter leg, the side lengths form a 30o-60o-90o triangle.

9.
The lengths of two legs are equal and hypotenuse is $\sqrt{2}$ 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. None of the above

#### Solution:

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.

The triangle is a 45o - 45o - 90o triangle.

10.
Find the measures of the missing sides in the triangle.

 a. AB = 6 cm and AC = 6$\sqrt{2}$ cm b. AB = 5 cm and AC = 8 cm c. AB = 6$\sqrt{2}$ cm and AC = 6 cm d. None of the above

#### Solution:

Since two of the angles of the triangle are equal and one of the angle is 90o, the triangle is an isosceles right triangle.

The legs have same length and hypotenuse is 2 times its leg in an isosceles right triangle.

AB, BC are legs and AC is the hypotenuse from the figure.

BC = AB = 6 cm
[Since legs have same lengths in an isosceles right triangle.]

AC = BC x 2 cm= 62 cm
[Substitute BC = 6.]

The measures of the missing sides, AB = 6 cm and AC = 62 cm.