Step: 1

[Apply Pythagorean theorem.]

Step: 2

[Subtract d ^{2} from both sides.]

Step: 3

[Substitute l and d .]

Step: 4

= 400 - 144

[Apply exponents and simplify.]

Step: 5

= 256

Step: 6

⇒ h = 256

[Take square root of both sides.]

Step: 7

= 16

Step: 8

Therefore, the height of the pole is 16 feet.

Correct Answer is : 16 feet

Step: 1

The triangle is a 45^{o}-45^{o}-90^{o} triangle.

Step: 2

Length of PR = 9√2 inches.

Step: 3

In a 45^{o}- 45^{o}-90^{o} 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 = PQ√2

Step: 6

PQ = PR/√2

[Divide each side by √2.]

Step: 7

PQ = 9√2/√2

[Replace PR with 9√2.]

Step: 8

PQ = 9 inches

[Simplify.]

Step: 9

Since the lengths of two legs are equal in 45^{o}-45^{o}-90^{o} triangle, PQ = QR = 9 inches.

Step: 10

The lengths of PQ and QR are 9 inches and 9 inches.

Correct Answer is : 9 inches and 9 inches

Step: 1

If one angle of the triangle is 90°, then the triangle is a right triangle.

Step: 2

The side opposite to right angle is hypotenuse.

Step: 3

Let x be the length of the third side of the triangle.

Step: 4

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

Step: 5

Applying Pythagorean theorem, 13^{2} = 12^{2} + x ^{2}

Step: 6

169 = 144 + x ^{2}

Step: 7

169 - 144 = x ^{2}

[Subtract 144 from both the sides.]

Step: 8

25 = x ^{2}

Step: 9

√25 = x

5 =x

5 =

Step: 10

The length of the third side of the triangle is 5 units.

Correct Answer is : 5 units

Step: 1

Let s be the side of the square garden and d be the distance Justin walked.

Step: 2

All the angles of a square are right angles. So, the diagonal of the square will be the hypotenuse of the right triangle formed by two adjacent sides and the diagonal.

Step: 3

[Apply Pythagorean theorem.]

Step: 4

[Substitute s = 16.]

Step: 5

[Apply exponents and simplify.]

Step: 6

[Take square root on both sides.]

Step: 7

The total distance Justin walked is 16.97 ft.

Correct Answer is : 16.97 ft

Step: 1

From the figure, c = 17 and b = 15

Step: 2

[Write Pythagorean theorem.]

Step: 3

[Subtract b ^{2} from both sides.]

Step: 4

[Substitute for b and c .]

Step: 5

[Apply exponents and simplify.]

Step: 6

[Take square root of both sides.]

Step: 7

The value of a is 8.

Correct Answer is : 8

Step: 1

The length of the ladder l = 10 feet.

Step: 2

The distance from the foot of the ladder to the wall, d = 6 feet.

Step: 3

Let h be the height of the wall.

Step: 4

[Write Pythagorean theorem.]

Step: 5

[Subtract d ^{2} from both sides.]

Step: 6

= 10^{2} - 6^{2}

[Substitute l and h .]

Step: 7

= 100 - 36

[Apply exponents and simplify.]

Step: 8

= 64

Step: 9

[Take square root of both sides.]

Step: 10

= 8

Step: 11

Height of the wall = 8 feet.

Correct Answer is : 8 feet

Step: 1

The lengths of hypotenuse and shorter leg of a right triangle are in the ratio 5:3

Step: 2

Let the length of hypotenuse and shorter leg be 5k and 3k respectively.

Step: 3

Let the length of longer leg be x .

Step: 4

[Apply Pythagorean theorem.]

Step: 5

[Evaluate powers.]

Step: 6

[Subtract and find the positive square root.]

Step: 7

The perimeter of triangle = 60 inches.

Step: 8

Perimeter of the right triangle = 5k + 4k + 3k = 12k .

Step: 9

12k = 60

Step: 10

[Divide both sides with 12 and simplify.]

Step: 11

The length of longer leg = 4k = 4 x 5 = 20 inches.

Correct Answer is : 20 inches

Step: 1

Let a be the length of the side and c be the length of hypotenuse.

Step: 2

Then, a = 3 ft and c = 4 ft.

Step: 3

Let b be the length of the side to be calculated.

Step: 4

According to Pythagorean theorem, hypotenuse^{2} = sum of the squares of other two sides.

Step: 5

Step: 6

4^{2} = 3^{2} + b ^{2}

[Substitute c = 4 and a = 3.]

Step: 7

16 = 9 + b ^{2}

Step: 8

[Subtract 9 from both the sides.]

Step: 9

Step: 10

Step: 11

The length of the other side = 7 ft.

Correct Answer is : 7 ft

Step: 1

Let s be the side of the square garden and d be the distance Jim walked.

Step: 2

All the angles of a square are right angles. So, the diagonal of the square will be the hypotenuse of the right triangle formed by two adjacent sides and the diagonal.

Step: 3

[Apply Pythagorean theorem.]

Step: 4

[Substitute s = 25.]

Step: 5

[Apply exponents and simplify.]

Step: 6

[Take square root on both sides.]

Step: 7

The total distance Jim walked is 35.35 ft.

Correct Answer is : 35.35 ft

Step: 1

Plot the points R(- 3, 2), S(- 3, - 2) and T(6, - 2) in a Cartesian plane and join them.

Step: 2

Distance between the point R and the point S = 2 - (-2) = 4 units

[Difference between the y -coordinates.]

Step: 3

Distance between the point S and the point T = 6 - (-3) = 9 units

[Difference between the x -coordinates.]

Step: 4

Distance between Andrew and Peter = Distance between the point R and the point T

Step: 5

Since ΔRST is a right-angled triangle,

RT^{2} = RS^{2} + ST^{2}

RT

[Pythagoras′ theorem.]

Step: 6

RT^{2} = 4^{2} + 9^{2}

[Substitute the values.]

Step: 7

RT^{2} = 16 + 81 = 97

[Add.]

Step: 8

RT = 9 7 = 9.85 units

[Take square root on both sides.]

Step: 9

Therefore, the distance between Andrew and Peter is 9.85 units.

Correct Answer is : 9.85 units

Step: 1

The radius of the circle OB = 5 in.

Step: 2

Diameter of the circle, BD = 10 in.

[diameter = 2 × radius.]

Step: 3

From right triangle BDC, BD^{2} = BC^{2} + CD^{2}

[Apply Pythagorean theorem.]

Step: 4

BD^{2} = BC^{2} + BC^{2}

[Since all sides of a square are equal,replace CD with BC.]

Step: 5

10^{2} = 2BC^{2}

[Substitute BD.]

Step: 6

100 = 2BC^{2}

[Simplify.]

Step: 7

BC^{2} = 100 2 = 50

[Divide by 2 on both sides.]

Step: 8

BC = 5 0 = 52

[Take square root of both sides.]

Step: 9

The length of side of square = 52 in.

Correct Answer is : 52 in.

Step: 1

The two legs of a right triangle are in the ratio 5:12

Step: 2

Let the length of two legs be 5k and 12k respectively.

Step: 3

Hypotenuse^{2} = (5k )^{2} + (12k )^{2}

[Apply Pythagorean theorem.]

Step: 4

Hypotenuse^{2} = 25k ^{2} + 144k ^{2}

[Apply exponents .]

Step: 5

= 169k ^{2}

[Simplify.]

Step: 6

Hypotenuse = 1 6 9 k 2 = 13k

[Take square root of both sides.]

Step: 7

The perimeter of right triangle = 210 inches.

Step: 8

Perimeter of the right triangle = 5k + 12k + 13k = 30k .

Step: 9

30k = 210

[Equate perimeters]

Step: 10

[Divide by 30 on both sides.]

Step: 11

Hypotenuse = 13k = 13 × 7 = 91

[Substitute k .]

Step: 12

The length of hypotenuse = 91 inches.

Correct Answer is : 91 inches

Step: 1

The lengths of hypotenuse and shorter leg of a right triangle are in the ratio 5:3

Step: 2

Let the length of hypotenuse and shorter leg be 5k and 3k respectively.

Step: 3

Let the length of longer leg be x .

Step: 4

[Apply Pythagorean theorem.]

Step: 5

[Evaluate powers.]

Step: 6

[Subtract and find the positive square root.]

Step: 7

The perimeter of triangle = 36 inches.

Step: 8

Perimeter of the right triangle = 5k + 4k + 3k = 12k .

Step: 9

12k = 36

Step: 10

[Divide both sides with 12 and simplify.]

Step: 11

The length of longer leg = 4k = 4 × 3 = 12 inches.

Correct Answer is : 12 inches

Step: 1

The diameter of the circle O = 34 cm = AB

Step: 2

Radius of the circle OA = OC = 34 2 = 17 cm

[Radius = Diameter 2 .]

Step: 3

OAC is a right triangle.

Step: 4

AC^{2} = OA^{2} + OC^{2}

[Apply Pythagorean theorem]

Step: 5

AC^{2} = (17)^{2} + (17)^{2}

[Replace OA and OC with 17.]

Step: 6

AC^{2} = 289 + 289 = 578

[Apply exponents and simplify.]

Step: 7

AC = 5 7 8 = 1 7 2

[Find the positive square root.]

Step: 8

The length of the chord AC = 172 cm.

Correct Answer is : 172 cm

- Parallel Lines and Transversals-Gr 8-Solved Examples
- Angle Sum Theorem of a Triangle-Gr 8-Solved Examples
- Exterior Angle Theorem-Gr 8-Solved Examples
- Distance Between Two Points-Gr 8-Solved Examples
- Transformations-Gr 8-Solved Examples
- Volume of Cylinders-Gr 8-Solved Examples
- Volume of Cones-Gr 8-Solved Examples
- Volume of a Sphere-Gr 8-Solved Examples

- Pythagorean Theorem