Question

124\. The hypotenuse of a right triangle is 10 inches. If one of the legs of the triangle is 8 inches long, what is the length of the other leg? (A) 2 inches (B) 6 inches (C) inches (D) 40 inches

Short answer

The length of the other leg is \(6\) inches.

Step-by-step solution

Write down the Pythagorean theorem formula

The formula for the Pythagorean theorem is: \[c^2 = a^2 + b^2\] where c is the length of the hypotenuse, and a and b are the lengths of the other two legs.

Substitute given values into the formula

We are given that the hypotenuse, c = 10 inches, and one of the legs, a = 8 inches. We need to find the length of the other leg, b. Substitute these values into the Pythagorean theorem formula: \[10^2 = 8^2 + b^2\]

Solve for the unknown leg length (b)

Now, we need to solve for b: \[100 = 64 + b^2\] Rearrange the equation to isolate b^2: \[b^2 = 100 - 64\] Subtract 64 from 100: \[b^2 = 36\] Take the square root of both sides of the equation: \[b = \sqrt{36}\] So, the length of the other leg, b, is: \[b = 6\] Now we can see that the answer is (B) 6 inches.

Key concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. This angle gives the right triangle its name. Because of this right angle, right triangles have unique properties that make them interesting and useful for solving various problems in geometry and trigonometry.

Key characteristics of a right triangle include:

• One angle measuring 90 degrees.
• The side opposite this right angle is called the hypotenuse, which is always both the longest side and situated opposite the right angle.
• The two sides that form the right angle are known as the legs of the triangle.

Understanding these basics can help you when applying the Pythagorean theorem to find the length of sides or angles in right triangles. This unique feature of having a right angle simplifies calculations and makes these triangles crucial in mathematical problem-solving.

Hypotenuse Calculation

Calculating the hypotenuse in a right triangle is a fundamental skill that uses the Pythagorean theorem. The Pythagorean theorem itself is essential, as it states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). This can be written as:\[ c^2 = a^2 + b^2 \]Where:

• \(c\) is the length of the hypotenuse.
• \(a\) and \(b\) are the lengths of the other two legs.

When you know the lengths of two sides, you can easily calculate the third side using this theorem. For example, if the hypotenuse \(c\) is 10 inches and one leg \(a\) is 8 inches, you would set up your equation as:\[ 10^2 = 8^2 + b^2 \]Solving this equation gives you the length of the unknown leg \(b\). This calculation is a core part of geometry problem-solving, especially when working with right triangles.

Geometry Problem-Solving

Geometry problem-solving often involves right triangles due to their unique and well-understood properties. Applying the Pythagorean theorem is one concrete way to address these problems effectively.

Here’s how to approach such problems:

• Identify the right triangle in your problem and confirm the known values. This often includes the hypotenuse and one of the legs.
• Use the Pythagorean theorem to set up equations involving the squares of these side lengths.
• Solve the equation step by step, remembering to isolate the variable when finding lengths of unknown sides.
• Check your solution by ensuring that it fits well within the context of the problem, confirming the hypotenuse is indeed the longest side.

Understanding these steps can increase your confidence in tackling a variety of problems that involve right triangles. This logical and stepwise problem-solving strategy reinforces strong geometry skills, useful for more complex mathematical challenges.