Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. \(a=7, b=4, c=?\)

Short answer

\(c \approx 8.1\)

Step-by-step solution

Identify the components

We have a right triangle with two legs, where \(a = 7\) and \(b = 4\). The hypotenuse \(c\) is unknown. Our goal is to find \(c\).

Use the Pythagorean Theorem

According to the Pythagorean Theorem, the relationship between the sides of a right triangle is \(a^2 + b^2 = c^2\). We will use this equation to find \(c\).

Substitute the known values into the equation

Substitute \(a = 7\) and \(b = 4\) into the theorem: \(7^2 + 4^2 = c^2\).

Calculate the squares of the known values

Calculate \(7^2 = 49\) and \(4^2 = 16\). So the equation becomes \(49 + 16 = c^2\).

Sum the squares and solve for \(c^2\)

Add the calculated squares: \(49 + 16 = 65\). So \(c^2 = 65\).

Solve for \(c\) by taking the square root

To find \(c\), take the square root of 65: \(c = \sqrt{65}\).

Round to the nearest tenth

Calculate \(\sqrt{65} \approx 8.0623\). Rounding to the nearest tenth, we get \(c \approx 8.1\).

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle is called the right angle. Understanding right triangles is essential because they form the basis of many geometric principles and calculations. In a right triangle, the side opposite the right angle is the longest side and is known as the hypotenuse. The other two sides, which are adjacent to the right angle, are referred to as the legs.

Right triangles are fascinating due to their unique properties:

• The sum of all interior angles is always 180 degrees.
• They play a crucial role in trigonometry, helping to define sine, cosine, and tangent functions.
• A famous application of right triangles is the Pythagorean Theorem, which provides a way to determine the length of the sides.

In many real-life scenarios, such as construction, carpentry, and computer graphics, right triangles are extensively used for precise measurements and calculations.

Hypotenuse Calculation

The hypotenuse is a key element in a right triangle as it is the longest side, lying directly opposite the right angle. Calculating the hypotenuse is an essential task tackled by the Pythagorean Theorem. This theorem establishes a relationship between the sides of a right triangle:

\[ a^2 + b^2 = c^2 \]

Where \( a \) and \( b \) are the lengths of the triangle's legs, and \( c \) is the length of the hypotenuse.

• To find the hypotenuse, we start by squaring the lengths of both legs.
• Next, we add these squared values together to get \( c^2 \).
• Finally, to solve for \( c \), we take the square root of the sum.

Calculating the hypotenuse proves to be a simple yet powerful geometric technique. It is often used in solving real-world problems where distances and lengths need to be measured accurately.

Square Root

The square root is a mathematical function that is used to determine what number, when multiplied by itself, gives the original number. In the context of the Pythagorean Theorem, the square root is crucial for solving for the hypotenuse \( c \). After obtaining \( c^2 \), the square root helps find the exact value of \( c \).

For example, if \( c^2 = 65 \),

• We compute \( c = \sqrt{65} \).
• Using a calculator, we find that \( \sqrt{65} \approx 8.0623 \).
• Rounding this result to the nearest tenth gives us \( c \approx 8.1 \).

The concept of taking a square root is not only applicable in geometry but is also widely used in different fields such as physics, engineering, and statistics for simplifying and solving complex equations.