Triangle Geometry Practice Test

Calculate the length of the hypotenuse of a right triangle with legs measuring 6 cm and 8 cm.

Answer explanation

Use the Pythagorean theorem: c² = a² + b². Here, a = 6 cm and b = 8 cm. So, c² = 6² + 8² = 36 + 64 = 100. Therefore, c = √100 = 10 cm. The hypotenuse is 10 cm.

A right triangle has one leg measuring 9 cm and the hypotenuse measuring 15 cm. Find the other leg.

Answer explanation

Use the Pythagorean theorem: a² + b² = c². Here, a = 9 cm, c = 15 cm. So, 9² + b² = 15². This gives 81 + b² = 225. Thus, b² = 144, and b = 12 cm. The other leg measures 12 cm.

The legs of a right triangle are equal, and the hypotenuse measures 10√2 cm. Find the length of each leg.

Answer explanation

In a right triangle with equal legs, let each leg be x. By the Pythagorean theorem: x^2 + x^2 = (10√2)^2. This simplifies to 2x^2 = 200, giving x^2 = 100, so x = 10 cm. Each leg measures 10 cm.

Find the area of a right triangle with a base of 5 cm and a height of 12 cm.

Answer explanation

The area of a right triangle is calculated using the formula: Area = 1/2 * base * height. Here, Area = 1/2 * 5 cm * 12 cm = 30 cm². Thus, the area of the triangle is 30 cm².

Express √50 in its simplest radical form.

Answer explanation

To simplify √50, factor it into √(25 * 2). Since √25 = 5, we have √50 = 5√2. Thus, the simplest radical form of √50 is 5√2.

Express √72 in its simplest radical form.

Answer explanation

To simplify √50, factor it into √(25 * 2). Since √25 = 5, we have √50 = 5√2. Thus, the simplest radical form of √50 is 5√2.

A ladder leans against a wall. The base of the ladder is 3 meters from the wall, and the ladder is 5 meters long. How high does the ladder reach?

Answer explanation

Using the Pythagorean theorem, where the ladder is the hypotenuse (5m), the distance from the wall is one leg (3m), we find the height (h) using h^2 + 3^2 = 5^2. Thus, h^2 = 25 - 9 = 16, so h = 4 meters.

A right triangle has legs measuring 11 cm and 60 cm. Find the length of the hypotenuse.

Answer explanation

Use the Pythagorean theorem: c² = a² + b². Here, a = 11 cm and b = 60 cm. Calculate c² = 11² + 60² = 121 + 3600 = 3721. Thus, c = √3721 = 61 cm. The hypotenuse is 61 cm.

Prove that a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle.

Answer explanation

To prove the triangle is a right triangle, we apply the Pythagorean theorem. Check if 7^2 + 24^2 = 25^2. Calculating, 49 + 576 = 625, which is true. Thus, the triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle.

Find the length of the diagonal of a rectangle with width 8 cm and height 15 cm.

Answer explanation

To find the diagonal of a rectangle, use the Pythagorean theorem: diagonal = √(width² + height²). Here, diagonal = √(8² + 15²) = √(64 + 225) = √289 = 17 cm.