Problem 59 - Prealgebra

Question

Compute the exact value of the given expression. \(\sqrt{5^{2}+12^{2}}\)

Step-by-Step Solution

Answer

The exact value of the expression is 13.

Step 1: Understand the Expression

The given expression is \( \sqrt{5^2 + 12^2} \). This resembles the format of the Pythagorean theorem, where you find the hypotenuse of a right triangle with legs of 5 and 12.

Step 2: Square Each Term Inside the Square Root

Calculate each square: \( 5^2 = 25 \) and \( 12^2 = 144 \). Now the expression inside the square root is \( 25 + 144 \).

Step 3: Add the Squares

Add the results of the squared terms: \( 25 + 144 = 169 \).

Step 4: Take the Square Root of the Sum

Now we need to find the square root of 169: \( \sqrt{169} = 13 \).

Key Concepts

Right Triangle

A right triangle is a special type of triangle which includes one angle that is exactly 90 degrees. The side opposite this right angle is always the longest side of the triangle called the hypotenuse. This distinct structure makes right triangles unique, and many geometric problems involve this type of triangle.

Key features of a right triangle include:

• One angle, exactly 90 degrees.
• The other two angles sum up to 90 degrees, ensuring the total for all three angles in the triangle is 180 degrees.
• The two shorter sides, known as legs, form the right angle.
• Their relationship to the hypotenuse is pivotal in calculations involving the Pythagorean theorem.

Understanding the nature of right triangles is fundamental when applying the Pythagorean theorem, which helps in finding the length of one side if the other two are known.

Hypotenuse

The hypotenuse is the most important side when you're solving problems involving right triangles. It's the side opposite the right angle and is the longest one.

This side is crucial in a variety of mathematical applications, particularly when using the Pythagorean theorem: \[ a^2 + b^2 = c^2 \]This formula signifies that when you square the lengths of the two legs and add them together, the result is equal to the square of the hypotenuse.

Why is the hypotenuse important?

• It's used to calculate distance in coordinate geometry and trigonometry.
• In real-world problems, it helps in finding diagonal distances like across a floor or field.
• Understanding its calculation through the Pythagorean theorem builds a foundation for more complex math or physics calculations.

Every time you solve problems involving right triangles, focusing on the hypotenuse quickly narrows down the solution by applying the theorem.

Square Root

The square root is a mathematical operation that finds the number which, when multiplied by itself, gives the original number. It’s denoted by the radical symbol \( \sqrt{} \).

Finding a square root is essential in solving many geometric problems, especially those involving the Pythagorean theorem.

Here's how square roots come into play in calculations:

• Solving equations involving squared terms, for example, finding the length of a side when given area.
• Used extensively in calculating distances, as it helps measure the hypotenuse in right triangles.
• In functions and calculus, it's applied to determine rates of change or optimization problems.

In the example given, taking the square root of 169 (since \( \sqrt{169} = 13 \)) was the last step in calculating the hypotenuse using the Pythagorean theorem. Understanding how to calculate and interpret square roots is foundational for mathematical literacy across many contexts.