Problem 72 - Algebra for College Students

Question

\(a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(c\) if \(a=3\) meters and \(b=7\) meters.

Step-by-Step Solution

Answer

\(c = \sqrt{58}\) meters.

Step 1: Identify the Formula

To find the length of the hypotenuse, we will use the Pythagorean theorem, which states \( a^2 + b^2 = c^2 \) where \( a \) and \( b \) are the lengths of the legs of a right triangle, and \( c \) is the length of the hypotenuse.

Step 2: Substitute the Known Values

Plug the given values \( a = 3 \) and \( b = 7 \) into the Pythagorean theorem. This gives us \( 3^2 + 7^2 = c^2 \).

Step 3: Calculate the Squares

Calculate the squares of \( a \) and \( b \). Since \( 3^2 = 9 \) and \( 7^2 = 49 \), substituting these values gives us \( 9 + 49 = c^2 \).

Step 4: Add the Squares

Add the squared values together: \( 9 + 49 = 58 \). Therefore, \( c^2 = 58 \).

Step 5: Solve for the Hypotenuse

To find \( c \), we take the square root of both sides. Thus, \( c = \sqrt{58} \). Since 58 cannot be simplified further, \( \sqrt{58} \) is the simplest radical form.

Key Concepts

Right Triangle

A right triangle is a special kind of triangle where one of the angles measures exactly 90 degrees. This angle is known as the right angle. The sides that form this right angle are called the legs of the triangle, and the side opposite the right angle is known as the hypotenuse. Right triangles have a unique property captured by the Pythagorean theorem, making them a key topic in geometry and trigonometry. Some key features of right triangles include:

• They have one right angle (90 degrees).
• The two sides forming the right angle are called legs.
• The longest side, opposite the right angle, is the hypotenuse.

Right triangles are not only foundational for understanding trigonometry but also appear in various real-world contexts, such as in architecture, navigation, and even in creating sports strategy. Understanding how to work with these triangles involves using the Pythagorean theorem, which allows us to solve for unknown side lengths.

Hypotenuse

The hypotenuse is the longest side of a right triangle. It's always opposite the right angle. To find the hypotenuse, we use the well-known Pythagorean theorem, which establishes a relationship between the hypotenuse and the legs of a right triangle. This relationship is given by:\[ a^2 + b^2 = c^2 \]Here, \( a \) and \( b \) are the leg lengths, and \( c \) represents the hypotenuse. This equation states that the sum of the squares of the two legs equals the square of the hypotenuse.In our exercise, given \( a = 3 \) meters and \( b = 7 \) meters, we substitute into the formula to calculate: \[ c^2 = 3^2 + 7^2 = 9 + 49 = 58 \]By solving \( c^2 = 58 \), we use a square root to find \( c \), making \( c \) equivalent to \( \sqrt{58} \). The hypotenuse is essential in several applications, from calculating distances to analyzing waveforms.

Simplest Radical Form

Simplest radical form is a way of expressing square roots as succinctly and cleanly as possible. It involves simplifying the square root into its lowest possible terms without any perfect square factors remaining under the radical.For example, when you find that \( c = \sqrt{58} \), you aim to express this in its simplest radical form. This step involves determining if the number under the square root, 58 in this case, has any square factors. Since 58 is not a perfect square and does not contain any smaller square factors within its divisors, \( \sqrt{58} \) is already in its simplest form.Simplifying radicals is vital as it provides the most reduced version of the expression while ensuring accuracy. When dealing with radical expressions:

• Check for perfect square factors for simplification.
• Ensure there are no fractions left within the radical.
• Express final answers in the most concise manner possible.

This method is crucial for making mathematical expressions more manageable and accessible, especially in further calculations or comparisons.