Problem 45 - Prealgebra

Question

How do the lengths of the sides of a right triangle relate to each other? Include an example of a set of numbers that represents the measures of the legs and hypotenuse of a right triangle.

Step-by-Step Solution

Answer

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides; for example, sides 3, 4, and 5.

Step 1: Understanding the Pythagorean Theorem

For a right triangle, the length of the sides relates through the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written algebraically as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

Step 2: Applying the Theorem to an Example

Let's consider an example with sides measuring 3 and 4 centimeters. We want to find the hypotenuse. Using the formula, \( c^2 = 3^2 + 4^2 \). Calculate each term: \( 3^2 = 9 \), \( 4^2 = 16 \). So, \( c^2 = 9 + 16 = 25 \).

Step 3: Solving for the Hypotenuse

To find the hypotenuse \( c \), take the square root of 25, since \( c^2 = 25 \). Therefore, \( c = \sqrt{25} \), which simplifies to \( c = 5 \). Thus, the sides 3, 4, and 5 form a right triangle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle characterized by one angle measuring exactly 90 degrees. This right angle is crucial because it defines the properties and behavior of the triangle. The sides of the right triangle have unique names and functions:

• The side opposite the right angle is known as the hypotenuse. It is always the longest side of the triangle.
• The other two sides that form the right angle are called the legs.

Understanding the structure of a right triangle is vital for applying the Pythagorean Theorem, which is central to mathematics education. In the context of this exercise, we used a right triangle with legs measuring 3 and 4 centimeters, and were able to find the hypotenuse using this foundational knowledge.

Hypotenuse

The hypotenuse in a right triangle holds a special place because it is always opposite the right angle. It is also the longest side. The importance of the hypotenuse is emphasized by the Pythagorean Theorem, which connects all three sides of the triangle.
In mathematical terms, if we have a right triangle with legs of lengths "a" and "b," and a hypotenuse of length "c," the Pythagorean Theorem tells us that:
- \( c^2 = a^2 + b^2 \)In our example problem, the hypotenuse was calculated by rearranging this formula to solve for "c." When the legs are 3 and 4, the hypotenuse was found to be 5. Understanding this relationship is a cornerstone of mathematics education.

Mathematics Education

Teaching the Pythagorean Theorem is a fundamental part of mathematics education because it provides insight into how geometry and algebra intersect. Students learn not only about spatial relationships but also acquire skills in:

• Problem-solving: applying a formula to find unknown values.
• Algebraic manipulation: rearranging equations to isolate a desired variable.
• Geometric reasoning: visualizing and working with shapes.

This exercise, featuring the calculation of hypotenuse length using values of 3 and 4, helps students practice these skills and see firsthand the power of this mathematical concept.

Algebra

Algebra plays a key role in understanding and applying the Pythagorean Theorem. It involves working with formulas and equations to solve for unknown variables. In our step-by-step solution:

• We expressed the relationship of the triangle’s sides using the equation \( c^2 = a^2 + b^2 \).
• We calculated the legs’ squares (e.g., \( 3^2 = 9 \)), showing proficiency in basic algebraic operations.
• We isolated the hypotenuse "c" by taking the square root of the summed squares.

These actions emphasize algebra's utility in mathematical problem-solving, allowing us to deduce unknown measurements from known quantities.

Problem-Solving

Problem-solving is central to applying the Pythagorean Theorem successfully. The process involves several well-defined steps:
First, correctly identifying which triangle sides are which is crucial, especially distinguishing between legs and the hypotenuse.
Next, apply the Pythagorean formula \( c^2 = a^2 + b^2 \) by substituting the known side lengths into the equation.
By calculating sums and performing operations like squaring and taking roots, you resolve the hypotenuse length.
This example showed how 3 and 4, as legs of the triangle, led us through basic calculations for a problem-solving experience that builds confidence in mathematics. Understanding and practicing these steps reinforce logical reasoning and analytical skills.