Problem 2 - Prealgebra

Question

Your solutions should include a well-labeled sketch. The length of one leg of a right triangle is 7 meters, and the length of the hypotenuse is 25 meters. Find the exact length of the other leg.

Step-by-Step Solution

Answer

The exact length of the other leg is 24 meters.

Step 1: Identify the formula

In a right triangle, the Pythagorean theorem is used, which is \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.

Step 2: Assign known values

We know one leg \( a = 7 \) meters and the hypotenuse \( c = 25 \) meters. We need to find the other leg \( b \).

Step 3: Substitute known values into the formula

Substitute the known values into the Pythagorean theorem: \( 7^2 + b^2 = 25^2 \).

Step 4: Simplify the equation

Calculate the squares: \( 49 + b^2 = 625 \).

Step 5: Solve for \( b \)

Rearrange the equation to solve for \( b^2 \): \( b^2 = 625 - 49 \). Therefore, \( b^2 = 576 \).

Step 6: Take the square root

Find \( b \) by taking the square root of 576: \( b = \sqrt{576} = 24 \).

Step 7: Sketch the triangle

Draw a right triangle and label the known side 7 meters, the hypotenuse 25 meters, and the other solved leg as 24 meters.

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This 90-degree angle is also known as a right angle. Right triangles are commonly found in geometry and are essential in various mathematical applications. They have unique properties that distinguish them from other types of triangles:

• One angle is a right angle, measuring 90 degrees.
• The side opposite to the right angle is the longest side, called the hypotenuse.
• The other two sides are known as the legs of the triangle.

The significance of right triangles lies in their ability to be analyzed using the Pythagorean theorem, which helps in calculating dimensions and solving problems involving space and distance. In the problem given, knowing you are dealing with a right triangle empowers you to apply this crucial theorem for a solution.

The Legs of a Triangle in a Right Triangle

In a right triangle, the two sides that form the right angle are called the legs. These legs are essential as they directly interact to create the 90-degree angle. Understanding and correctly identifying these legs is crucial because these are the sides you use in the Pythagorean theorem. Typically, we label these legs as \(a\) and \(b\).

• The lengths of these legs vary based on the triangle's orientation and size, but together they determine the hypotenuse's magnitude.
• In problems, you might know the length of one leg and need to find the other, as illustrated in our problem where one leg is 7 meters, and we found the other to be 24 meters.

Proper comprehension of the relationship between the legs is vital for solving geometric problems and understanding the spatial layout of the right triangle.

Intricacies of the Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. Due to its position and length, the hypotenuse holds a key position in calculations and geometric principles, as demonstrated in the Pythagorean theorem.

• In our exercise, the hypotenuse is given as 25 meters, setting a constraint necessary for calculating the unknown leg.
• To correctly apply the Pythagorean theorem \(a^2 + b^2 = c^2\), the hypotenuse is always represented by \(c\).

The accuracy in identifying and using the hypotenuse ensures problems are correctly solved, reinforcing the side's importance in right triangle equations and physics applications.