Problem 78 - 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. $$a=7, b=24 ; \text { find } c$$

Step-by-Step Solution

Answer

The length of the hypotenuse is 25.

Step 1: Recall the Pythagorean Theorem

In a right triangle, the Pythagorean Theorem relates the lengths of the two legs, denoted as \(a\) and \(b\), and the hypotenuse, denoted as \(c\). The theorem states that \(a^2 + b^2 = c^2\).

Step 2: Insert Known Values

Substitute the given values into the Pythagorean Theorem: \(a = 7\) and \(b = 24\). Thus, the equation becomes \(7^2 + 24^2 = c^2\).

Step 3: Calculate Squares of Known Values

Calculate the squares of \(7\) and \(24\). This results in \(7^2 = 49\) and \(24^2 = 576\).

Step 4: Add the Squares

Add the results from Step 3: \(49 + 576 = 625\). So, \(c^2 = 625\).

Step 5: Solve for the Hypotenuse

To find \(c\), take the square root of 625. That is \(c = \sqrt{625} = 25\). Therefore, the length of the hypotenuse is 25.

Key Concepts

Right Triangle

A right triangle is a type of triangle that includes one angle specifically measuring 90 degrees. This angle is known as a right angle. The concept of right triangles is foundational in geometry, particularly because they exhibit unique properties not shared by other triangles. In such a triangle, the sides are generally referred to as:

• The two shorter sides, each forming an angle with the right angle, are known as the legs.
• The longest side, which lies opposite the right angle, is called the hypotenuse.

Understanding right triangles is critical because they allow us to use the Pythagorean Theorem, opening up opportunities to solve practical problems that involve measurement and distance.

Hypotenuse

The hypotenuse is uniquely significant in a right triangle, being the longest side and directly opposite the right angle. It serves as a key variable in the Pythagorean Theorem, which is a fundamental rule in mathematical geometry. This theorem allows us to link the hypotenuse with the other two sides, which we call the legs, by establishing the relationship: \[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse and \( a \) and \( b \) are the two legs. This relationship underlines that the square of the hypotenuse equals the sum of the squares of the other two sides.
By understanding the role of the hypotenuse, students can solve problems relating to distances and heights in both theoretical and real-world contexts.

Square Root

The notion of a square root is indispensable when working with the Pythagorean Theorem. After expressing the sum of the squares of the legs as \( c^2 \), finding \( c \) involves determining the square root of that value. The square root of a number \( x \) is another number which, when multiplied by itself, gives \( x \). This is denoted as \( \sqrt{x} \). In our exercise, once we calculated \( c^2 = 625 \), we used the square root to find the hypotenuse: \[ c = \sqrt{625} = 25 \] This operation turns what might seem an abstract computation into a concrete answer, ultimately solving for the unknown side in a right triangle.

Geometry

Geometry is a vast field of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. Right triangles, along with the Pythagorean Theorem, are some of the cornerstones of this area. In studying right triangles, we specifically delve into planar geometry, which deals with shapes like triangles that lie within a flat plane. This branch provides essential tools for understanding space, shapes, and constructive proofs.
Applying geometric principles allows for the calculation of distances and angles, making it highly applicable in fields like engineering, architecture, and physical sciences.
By mastering these basic geometric concepts, students can extend their capabilities to analyze more complex forms, fostering a deeper appreciation of mathematical balance and design.