Problem 33 - Prealgebra

Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. $$a=\sqrt{177}=, b=?, c=31$$

Step-by-Step Solution

Answer

The missing measure \(b\) is 28.

Step 1: Identify the Known Values and Equation

We are given that in a right triangle, the hypotenuse \(c\) is 31, and one leg \(a\) is \(\sqrt{177}\). We need to find the missing leg \(b\). In a right triangle, according to the Pythagorean Theorem, the equation is \(a^2 + b^2 = c^2\).

Step 2: Substitute Known Values into the Equation

Substitute \(a = \sqrt{177}\) and \(c = 31\) into the equation. Thus, it becomes \((\sqrt{177})^2 + b^2 = 31^2\).

Step 3: Simplify the Equation

Calculate \( (\sqrt{177})^2 \), which simply returns 177. The equation transforms to \(177 + b^2 = 961\), since \(31^2 = 961\).

Step 4: Solve for \(b^2\)

Subtract 177 from both sides of the equation to isolate \(b^2\). This gives \(b^2 = 961 - 177\).

Step 5: Calculate \(b^2\)

Compute \(961 - 177\), which equals 784. So, \(b^2 = 784\).

Step 6: Find \(b\) by Taking the Square Root

To find \(b\), take the square root of 784. Thus, \(b = \sqrt{784}\).

Step 7: Simplify the Square Root

Calculate \(\sqrt{784}\), which equals 28. Thus, \(b = 28\).

Key Concepts

Right Triangle

A right triangle is a triangle that has one angle exactly equal to 90 degrees. This unique feature makes the right triangle very special in geometry, as it has certain properties that do not apply to other types of triangles.
The sides of a right triangle have specific names:

• The side opposite the right angle is called the hypotenuse. This is always the longest side.
• The other two sides, which are adjacent to the right angle, are called legs.

In any right triangle, the Pythagorean Theorem holds true. This theorem is crucial for finding the lengths of the sides when we know at least two of them. Understanding right triangles is essential in solving many real-world problems, from construction to navigation.

Hypotenuse

The hypotenuse is the longest side of the right triangle, lying directly opposite the 90-degree angle. In the Pythagorean Theorem, the hypotenuse is represented by the letter c in the equation a² + b² = c².
When dealing with practical problems and exercises, knowing the hypotenuse can greatly simplify finding unknown measures, as it typically allows solving the triangle's geometry with only basic calculations. The hypotenuse is central to the understanding of triangle relationships, helping us connect theoretical geometry with practical applications.

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. For example, the square root of 177 is a number which, when squared, results in 177. This is written as \(\sqrt{177}\).
In the exercise, finding the square root helps us revert from a squared value back to its original form. This technique is useful not only in solving geometry problems like finding triangle sides but also in various fields like physics or engineering where we frequently deal with squared values.
Using the square root can often seem challenging at first. However, recognizing its role in reversing squared calculations makes it a powerful tool in mathematical problem-solving.

Geometry

Geometry is the branch of mathematics that studies the sizes, shapes, and properties of figures and spaces. Understanding geometry is crucial when solving problems involving shapes, especially triangles.
In this problem, the Pythagorean Theorem is a geometric principle that applies specifically to right triangles, allowing us to find side lengths using algebraic methods.

• Geometry helps in visualizing and structuring problems.
• It connects abstract mathematical calculations to real-world spatial arrangements.

By mastering geometry, students can enhance their problem-solving skills, making it easier to tackle various challenges involving spatial reasoning and dimensional analysis.