Problem 13 - Algebra 1: Concepts and Skills

Question

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ a=3, c=4 $$

Step-by-Step Solution

Answer

The length of the side \(b\) is approximately 2.65.

Step 1: Applying the Pythagorean theorem

The Pythagorean theorem forms the basis for solving this problem. The theorem states that in any right triangle, 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 expressed as \(c^2 = a^2 + b^2\). Given \(a = 3\) and \(c = 4\), we can substitute these values into the equation and solve for \(b\).

Step 2: Substitute a and c into the equation

Substitute \(a = 3\) and \(c = 4\) into the Pythagorean theorem. So, \(4^2 = 3^2 + b^2\), which simplifies to \(16 = 9 + b^2\).

Step 3: Solving for b

Solving the equation \(16 = 9 + b^2\) for \(b\), we subtract 9 from both sides to isolate \(b^2\) on one side. The equation becomes \(b^2 = 16 - 9\), which simplifies to \(b^2 = 7\). Calculate the square root of 7 to find the length of \(b\). Doing this we find that \(b \approx 2.65\).

Key Concepts

Right Triangle

A right triangle is a unique type of triangle that has one angle measuring exactly 90 degrees. This angle is known as the right angle, and it sets apart right triangles from other types of triangles. Right triangles are commonly found in architecture, design, and everyday objects like ramps and staircases.

Within a right triangle, there are three sides:

• Two legs, which are the shorter sides.
• The hypotenuse, which is the longest side opposite the right angle.

These triangles have special properties that make them useful for calculations, especially with the Pythagorean Theorem.

Understanding right triangles helps us in a range of mathematical calculations, especially when dealing with angles and lengths in two-dimensional spaces.

Hypotenuse

The hypotenuse is a key element of a right triangle. It is the longest side and always lies opposite the right angle. Understanding the role of the hypotenuse is crucial when using the Pythagorean Theorem.

Here are some important points about the hypotenuse:

• In any right triangle, the hypotenuse is longer than either leg.
• The length of the hypotenuse plays a central role in calculating other dimensions of the triangle.

For instance, in our given problem, the hypotenuse has a length of 4. Recognizing it helps us determine other missing side lengths. By knowing two side lengths, the third can be found with ease using the theorem.

Solving Equations

Solving equations is a fundamental skill in math, and is particularly essential when working with the Pythagorean Theorem. It involves handling mathematical expressions to find the value of unknown variables.

Let's break down the process from our exercise:

• Start with the Pythagorean theorem: the formula is \(c^2 = a^2 + b^2\).
• Substitute the known values into the equation: \(4^2 = 3^2 + b^2\).
• Simplify and rearrange the equation: calculate \(4^2 = 16\) and \(3^2 = 9\), resulting in \(16 = 9 + b^2\).
• Isolate \(b^2\) by subtracting: \(16 - 9 = b^2\).
• Find \(b\) by taking the square root of 7, which gives \(b \approx 2.65\).

Each of these steps requires attention to detail and ensures that the calculations lead to an accurate determination of the unknown side.