Question

How long is the hypotenuse of a right triangle 24 square inches in area and with one leg 6 inches long? A. 24 inches B. 18 inches C. 12 inches D. 10 inches

Short answer

The hypotenuse is 10 inches long.

Step-by-step solution

Calculate the other leg using area formula

The area of a right triangle is given by \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). We are given the area (24 square inches) and one leg (6 inches). Set up the equation: \(\frac{1}{2} \times 6 \times \text{other leg} = 24\). Solving for the other leg gives \(\text{other leg} = \frac{24 \times 2}{6} = 8 \text{ inches}\).

Apply the Pythagorean Theorem

With the legs of the right triangle known (6 inches and 8 inches), use the Pythagorean Theorem to find the hypotenuse. The theorem states: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. Substitute the leg lengths: \(6^2 + 8^2 = c^2\).

Solve for the hypotenuse

Calculate the squares of the legs: \(6^2 = 36\) and \(8^2 = 64\). Then, sum these values: \36 + 64 = 100\, so \c^2 = 100\. Taking the square root of both sides, \c = \sqrt{100} = 10 \text{ inches}\.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It’s used to determine the length of the sides in a right triangle. A right triangle features one 90-degree angle. This theorem helps us find the hypotenuse, which is the side opposite this right angle. The theorem states:

• Formula:
• Application:

The equation is written as: \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the triangle’s legs and \( c \) is the hypotenuse.

Triangle Area Calculation

To solve problems involving triangle measurements, understanding how to calculate the area is essential. Specifically, in a right triangle, the formula involves the base and the height, which are the two legs that form the right angle.

• Formula: The area of a right triangle is given by \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
• Example: For a triangle with a base of 6 inches and an unknown height, if the area is 24 square inches, the equation becomes \( \frac{1}{2} \times 6 \times \text{height} = 24 \).
• Simplifying: Solving the equation, we find the other leg to be 8 inches since \( \text{height} = \frac{24 \times 2}{6} = 8 \text{ inches} \).

By mastering this, you can easily handle related questions, making GED math problems straightforward.

GED Math Problems

GED math problems often include questions about right triangles. These problems assess your understanding of the Pythagorean Theorem and area calculations. Here's a breakdown of how to handle such problems:

By following a structured approach, as done in the original solution, you can tackle these problems confidently and accurately.