Canada Grade 8 Math The Pythagorean theorem

A right-angled triangle has one leg measuring 3 cm and the other leg measuring 4 cm. What is the length of the hypotenuse?

• 5 cm

• 6 cm

• 7 cm

• 8 cm

Answer explanation

To find the hypotenuse of a right-angled triangle, use the Pythagorean theorem: a² + b² = c². Here, 3² + 4² = 9 + 16 = 25. Thus, c = √25 = 5 cm. Therefore, the hypotenuse is 5 cm.

In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. What is the length of the other leg?

• 10 cm

• 12 cm

• 11 cm

• 9 cm

Answer explanation

Using the Pythagorean theorem: a² + b² = c². Here, c = 13 cm and one leg (a) = 5 cm. So, 5² + b² = 13². This simplifies to 25 + b² = 169, leading to b² = 144. Thus, b = 12 cm, which is the length of the other leg.

If a right-angled triangle has legs of lengths 6 cm and 8 cm, what is the area of the triangle?

• 24 cm²

• 48 cm²

• 36 cm²

• 30 cm²

Answer explanation

The area of a right-angled triangle is calculated using the formula: Area = (1/2) * base * height. Here, the legs are 6 cm and 8 cm, so Area = (1/2) * 6 * 8 = 24 cm². Thus, the correct answer is 24 cm².

A ladder is leaning against a wall. The foot of the ladder is 9 m away from the wall, and the ladder reaches a height of 12 m on the wall. How long is the ladder?

• 15 m

• 16 m

• 13 m

• 14 m

Answer explanation

To find the length of the ladder, use the Pythagorean theorem: a^2 + b^2 = c^2. Here, a = 9 m (distance from the wall), b = 12 m (height), so c = √(9^2 + 12^2) = √(81 + 144) = √225 = 15 m.

A square has a diagonal of 10 cm. What is the length of each side of the square?

• 5 cm

• 7 cm

Answer explanation

To find the side length of a square with a diagonal of 10 cm, use the formula: diagonal = side × √2. Thus, side = diagonal / √2 = 10 / √2 = 10√2 / 2 = √50 cm. Therefore, the correct answer is √50 cm.

The sides of a right-angled triangle are in the ratio 3:4:5. If the hypotenuse is 20 cm, what is the length of the shortest side?

• 12 cm

• 15 cm

• 16 cm

• 18 cm

Answer explanation

The sides of the triangle are in the ratio 3:4:5. If the hypotenuse (5 parts) is 20 cm, each part is 4 cm. The shortest side (3 parts) is 3 * 4 = 12 cm.

A right-angled triangle has a hypotenuse of 17 cm and one leg of 8 cm. What is the length of the other leg?

• 15 cm

• 14 cm

• 13 cm

• 12 cm

Answer explanation

Using the Pythagorean theorem, a² + b² = c², where c is the hypotenuse. Here, 8² + b² = 17². This simplifies to 64 + b² = 289. Thus, b² = 225, giving b = 15 cm. Therefore, the length of the other leg is 15 cm.

In a right-angled triangle, the legs are 9 cm and 12 cm. What is the length of the hypotenuse?

• 15 cm

• 16 cm

• 18 cm

• 20 cm

Answer explanation

To find the hypotenuse in a right-angled triangle, use the Pythagorean theorem: c² = a² + b². Here, c² = 9² + 12² = 81 + 144 = 225. Thus, c = √225 = 15 cm. Therefore, the hypotenuse is 15 cm.

A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Is this a right-angled triangle?

• Yes

• No

Answer explanation

To determine if the triangle is right-angled, we can use the Pythagorean theorem. Here, 7^2 + 24^2 = 49 + 576 = 625, which equals 25^2. Since this holds true, the triangle is a right-angled triangle. Thus, the answer is Yes.

The hypotenuse of a right-angled triangle is 26 cm, and one of the legs is 10 cm. What is the length of the other leg?

• 24 cm

• 20 cm

• 22 cm

• 18 cm

Answer explanation

Using the Pythagorean theorem: a² + b² = c². Here, c = 26 cm and one leg (a) = 10 cm. So, 10² + b² = 26². This simplifies to 100 + b² = 676, leading to b² = 576. Thus, b = 24 cm.