Problem 83 - A Graphical Approach to College Algebra

Question

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$b=\sqrt{13}, c=\sqrt{29} ; \text { find } a$$

Step-by-Step Solution

Answer

The length of side a is 4.

Step 1: Recall the Pythagorean Theorem

The Pythagorean Theorem states that in a right-angled triangle, the sum of the squares of the two legs (a and b) equals the square of the hypotenuse (c). This can be written as: \( a^2 + b^2 = c^2 \).

Step 2: Substitute Known Values

Insert the given values into the Pythagorean Theorem formula: \( a^2 + (\sqrt{13})^2 = (\sqrt{29})^2 \).

Step 3: Simplify the Equation

Calculate the squares of the known side lengths: \( a^2 + 13 = 29 \) because \((\sqrt{13})^2 = 13\) and \((\sqrt{29})^2 = 29\).

Step 4: Solve for a^2

Isolate \( a^2 \) by subtracting 13 from both sides of the equation: \( a^2 = 29 - 13 \).

Step 5: Simplify the Right-Hand Side

Simplify the equation: \( a^2 = 16 \).

Step 6: Solve for a

Take the square root of both sides to solve for \( a \): \( a = \sqrt{16} \).

Step 7: Find the Value of a

Calculate the square root: \( a = 4 \).

Key Concepts

Understanding Right Triangles

A right triangle is one of the basic shapes in geometry that consists of three sides: two shorter sides called 'legs' and one longer side called the 'hypotenuse'. The key characteristic of a right triangle is that it has one angle that is exactly 90 degrees. This right angle creates an important mathematical relationship between the sides, which we can use to solve problems involving right triangles. The sides adjacent to the right angle are the legs, while the side opposite the right angle is the hypotenuse, and is the longest side. In the Pythagorean Theorem, the two legs are usually denoted as \(a\) and \(b\), and the hypotenuse is denoted as \(c\).

For any right triangle:

• The legs define the height and base of the triangle.
• The hypotenuse is always opposite the right angle.

This understanding leads naturally into the use of the Pythagorean Theorem, which directly connects the lengths of all three sides.

Solving for Side Length

The process of solving for a side length in a right triangle typically involves using the Pythagorean Theorem. This theorem provides a formulaic way to determine any side when the other two are known. The general formula is \( a^2 + b^2 = c^2 \), where \(c\) is the hypotenuse.

In exercises like this, the goal is to find the missing side of the triangle when two other sides are known. For instance:

• Given \(b=\sqrt{13}\) and \(c=\sqrt{29}\), we want to find \(a\). Inserting these values into the formula looks like: \(a^2 + (\sqrt{13})^2 = (\sqrt{29})^2\).
• Solve for \(a\) by rearranging the formula to \(a^2 = \text{hypotenuse}^2 - \text{leg}^2\). That gives you \(a^2 = 29 - 13\).
• Simplify this to find \(a^2 = 16\).

Once you have \(a^2\), you solve for \(a\) by taking the square root, which leads us to our next crucial topic.

Square Root Calculations Explained

Square root calculations are essential when working with the Pythagorean Theorem, especially when determining the lengths of sides. Taking the square root of a number essentially asks what number multiplied by itself results in the given number. For example, the square root of 16 is 4, because \(4 \times 4 = 16\). Calculating square roots is straightforward, and it is a crucial skill for solving equations like \(a^2 = 16\).

Here's how you can approach square roots:

• Identify the square \(a^2\) you need to solve; in this case, \(a^2 = 16\).
• Find the number that when multiplied by itself equals 16. Here, \(\sqrt{16} = 4\).
• Use the result to determine the other side length of the triangle, and ensure you interpret all square root results positively in the context of triangle side lengths.
  
  

Remember that square root calculations are not just about finding numbers, but making connections between geometry and algebra to understand real-world scenarios, like the dimensions in a right triangle.