Problem 4 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. $$a=?, b=24, c=25$$

Step-by-Step Solution

Answer

The missing measure \(a\) is 7.

Step 1: Identify the Right Triangle

Recognize this as a right triangle problem where we know one leg, \(b = 24\), and the hypotenuse \(c = 25\). We need to find the missing leg \(a\).

Step 2: Recall the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle: \[ a^2 + b^2 = c^2 \] where \(a\), \(b\), and \(c\) are the lengths of the two legs and the hypotenuse, respectively.

Step 3: Substitute Known Values

Substitute the known values \(b = 24\) and \(c = 25\) into the Pythagorean Theorem:\[ a^2 + 24^2 = 25^2 \]

Step 4: Calculate Squares

Calculate the squares:\(24^2 = 576\) and \(25^2 = 625\). Substitute these into the equation:\[ a^2 + 576 = 625 \]

Step 5: Solve the Equation for \(a\)

Isolate \(a^2\) by subtracting 576 from both sides:\[ a^2 = 625 - 576 \]\[ a^2 = 49 \]

Step 6: Determine \(a\)

Since \(a^2 = 49\), take the square root of both sides to find \(a\):\[ a = \sqrt{49} = 7 \]

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle where one of the angles is exactly 90 degrees. This special angle is known as a 'right angle'. Right triangles are unique because of this angle, which allows for certain mathematical relationships to be used, such as the Pythagorean Theorem. Right triangles have three sides:

• The two shorter sides are called 'legs'.
• The longest side, opposite the right angle, is called the 'hypotenuse'.

In our problem, we are given a right triangle where the hypotenuse is 25, and one of the legs is 24. The task is to find the length of the other leg. Understanding the layout and properties of a right triangle is crucial for effectively applying theorems to solve problems like these.

Discovering the Hypotenuse

The hypotenuse is a key component in the structure of every right triangle. It is always the longest side, resting opposite the right angle. In the context of the Pythagorean Theorem, the hypotenuse is always labeled as 'c'. This side is crucial because the theorem essentially revolves around calculating or using its length.

• In the theorem, the equation is written as: \[ a^2 + b^2 = c^2 \]
• This format places the hypotenuse separately, highlighting its importance in these geometric calculations.

In our exercise, the hypotenuse is given as 25, which allows us to easily set up the equation to find the missing side with the known side lengths. Having the hypotenuse length provided simplifies the problem-solving process significantly.

Steps in Mathematical Problem Solving

Solving a problem like finding a missing length in a right triangle involves following structured steps. This methodical approach ensures accuracy and understanding. Let's breakdown the steps:

• Identification: Recognize that you're dealing with a right triangle and identify the known and unknown components.
• Theorem Application: Use the Pythagorean Theorem, \[ a^2 + b^2 = c^2 \], which relates the sides of a right triangle.
• Substitution: Insert known values into the theorem, which in our case are \( b = 24 \) and \( c = 25 \).
• Calculation: Compute the squares of these numbers to solve the equation.
• Solving: Isolate the variable by algebraic manipulation and solve for the unknown, which involves simple arithmetic.
• Verification: Ensure the solution is reasonable, especially when approximate values are involved, by checking the calculation steps.

By following these steps, you're not only solving the immediate problem but also reinforcing your mathematical reasoning and problem-solving skills. This structured approach is applicable to numerous mathematical challenges, providing a solid foundation for further learning.