Problem 11 - Prealgebra

Question

Your solutions should include a well-labeled sketch. The lengths of two legs of a right triangle are 6 meters and 8 meters. Find the exact length of the hypotenuse.

Step-by-Step Solution

Answer

The hypotenuse is 10 meters.

Step 1: Understand the Problem

You need to find the hypotenuse of a right triangle where the two legs have lengths of 6 meters and 8 meters. The hypotenuse is the side opposite the right angle and is the longest side of the triangle.

Step 2: Use the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). Therefore, you need to apply the formula: \[c^2 = a^2 + b^2\]. In this problem, let \(a = 6\) and \(b = 8\).

Step 3: Substitute into the Formula

Substitute the known values into the Pythagorean Theorem formula: \[c^2 = 6^2 + 8^2\]. Calculate the squares of the numbers: \[6^2 = 36\] and \[8^2 = 64\].

Step 4: Add the Squares of the Legs

Add the squares you calculated: \[36 + 64 = 100\]. This result gives you \(c^2 = 100\).

Step 5: Solve for the Hypotenuse

To find \(c\), take the square root of both sides: \(c = \sqrt{100}\). Calculate the square root: \(c = 10\).

Step 6: Sketch the Triangle

Draw a right triangle with the two legs labeled as 6 meters and 8 meters. Label the hypotenuse as 10 meters. This visual will help confirm your solution is consistent with the problem statement.

Key Concepts

Right Triangle

Understanding what a right triangle is can help demystify many geometric problems. A right triangle is a special type of triangle where one of the angles is exactly 90 degrees.
When you have a right triangle, it is often depicted with a small square symbol at the right angle to highlight its presence. A unique feature of right triangles is that they allow the usage of the Pythagorean Theorem to determine the relationship between their sides:

• One of the angles is always 90 degrees.
• The two sides forming the right angle are called 'legs'.
• The side opposite the 90-degree angle is known as the 'hypotenuse'.

In solving geometric problems involving right triangles, you will frequently be required to find the length of a side using given information about the other two sides. This will most likely involve the systematic application of the Pythagorean Theorem.

Hypotenuse

The hypotenuse is a critical component of any right triangle. It's the longest side of the triangle and opposite the right angle.
When applying the Pythagorean Theorem, the length of the hypotenuse is often the central focus since it can be derived if the lengths of the other two sides are known.To find the hypotenuse using the Pythagorean Theorem, you follow these steps:

• Square the lengths of both legs of the triangle.
• Add these squared values together.
• Take the square root of this sum to find the hypotenuse.

Using the example of a right triangle with legs measuring 6 meters and 8 meters, substituting into the Pythagorean formula gives: \[c^2 = 6^2 + 8^2\] Calculating gives \[c^2 = 36 + 64 = 100\] Taking the square root results in \(c = 10\) meters, confirming the hypotenuse's length.

Geometry

Geometry is more than just shapes and figures; it’s about understanding their properties and relationships.
When studying right triangles, geometry provides the tools and formulas to determine side lengths, angle measures, and more. Here’s how geometry comes into play with right triangles and the Pythagorean Theorem:

• Use geometry to label each part of the triangle. This helps you visualize and solve problems efficiently.
• Geometric principles enhance comprehension when applying the Pythagorean Theorem to find the hypotenuse or a missing leg length.
• Drawing diagrams helps verify your calculations and conclusions, ensuring they align with spatial realities.

Incorporating these geometric strategies, you confirm answers more confidently. Ensuring all parts of the triangle, such as angles and sides, fit together makes geometry a critical aspect of solving right triangle problems.