Question

The lengths of the legs of a right triangle are 4 inches and 5 inches, respectively. What is the length of the hypotenuse? (A) 3 inches (B) 9 inches (C) inches (D) 41 inches

Short answer

The length of the hypotenuse is \(\sqrt{41}\) inches.

Step-by-step solution

Identify the given information and the Pythagorean theorem

We are given the lengths of the legs of a right triangle (4 inches and 5 inches) and need to find the length of the hypotenuse. The Pythagorean theorem tells us that for a right triangle with sides a, b and hypotenuse c, \(a^2 + b^2 = c^2\).

Substitute the given lengths into the Pythagorean theorem

Replace a with the length of the first leg (4 inches) and b with the length of the second leg (5 inches). We get: \((4)^2 + (5)^2 = c^2\).

Calculate the sum of the square of the lengths

Now we need to calculate the sum of the square of the lengths: \((4)^2 + (5)^2 = 16 + 25 = 41\).

Solve for the length of the hypotenuse

Since we have the sum of the square of the lengths, we just need to find the square root to get the length of the hypotenuse: \(c = \sqrt{41}\). The correct answer is (D) 41 inches.

Key concepts

Understanding the Right Triangle

A right triangle is a type of triangle with one angle measuring exactly 90 degrees. This special right angle forms the cornerstone of right triangles and is key in recognizing them. The three sides of a right triangle have specific names: the two sides that form the right angle are called "legs," and the longest side opposite the right angle is known as the "hypotenuse".

Right triangles are unique because they adhere to the Pythagorean theorem, a fundamental rule in geometry that helps us solve various mathematical problems. It's important because every time you have a right triangle situation, you can apply this theorem to discover unknown side lengths as long as you know at least two sides.

The right triangle we are discussing is special because the lengths of the legs are given. This makes it convenient to apply the Pythagorean theorem and find the missing hypotenuse. Knowing how to identify and label the sides of a right triangle sets the foundation for solving related mathematical problems efficiently. Here, the given leg lengths are 4 inches and 5 inches.

Hypotenuse Calculation with the Pythagorean Theorem

The hypotenuse is the longest side of a right triangle, lying opposite the right angle. Calculating this side can often seem daunting, but with the Pythagorean theorem, it becomes a straightforward task.

The Pythagorean theorem states:

• Formula: For a right triangle with legs of length 'a' and 'b' and hypotenuse 'c', the theorem is: \[ a^2 + b^2 = c^2 \]

To find the hypotenuse:

• Substitute the known values: Here, substitute 4 and 5 for 'a' and 'b', respectively.
• Calculate the squares: For this triangle, it becomes \[ (4)^2 + (5)^2 = c^2 \].
• Sum the squares: Calculate \[ 16 + 25 = 41 \].
• Find the square root: Finally, calculate the square root to find \( c = \sqrt{41} \).

The hypotenuse of this right triangle is therefore \( \sqrt{41} \), an irrational number that you can approximate if needed. The ease with which you solve these calculations comes with practice and understanding of the underlying geometric principles.

Approaching Mathematical Problem-Solving

Mathematical problem-solving can sometimes seem intimidating, but it's all about breaking down the problem into manageable parts and applying known formulas and principles.

When solving a problem like finding the hypotenuse of a right triangle, you start by understanding the problem context and identifying what is given and what is required. This process is crucial:

• Understand the Problem: Ensure you fully grasp the question; here, it's calculating a hypotenuse.
• Identify the Formula: Recognize that the Pythagorean theorem is needed for a right triangle.
• Substitute the Values: Insert known measurements into the formula.
• Compute Step-by-step: Perform the calculations such as squaring numbers and solving equations.
• Check Your Work: Verify calculations make sense with the context.

This structured approach helps in reducing errors and enhances understanding of mathematical concepts. Practicing this method across various problems not only aids in specific cases like right triangles but also builds confidence in handling more complex math problems effectively.