Question

If \(c\) is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. \(a=\sqrt{7}, b=\sqrt{9}, c=?\)

Short answer

The hypotenuse \(c\) measures 4 units.

Step-by-step solution

Understand the Problem

You are given two sides of a right triangle: \(a = \sqrt{7}\) and \(b = \sqrt{9}\), which is equivalent to \(b = 3\). You need to find the length of the hypotenuse \(c\).

Use the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, \(c^2 = a^2 + b^2\). Substitute the given values into this equation: \(c^2 = (\sqrt{7})^2 + 3^2\).

Calculate the Squared Values

Now calculate the squared values: \((\sqrt{7})^2 = 7\) and \(3^2 = 9\). Thus, the equation becomes \(c^2 = 7 + 9\).

Solve for c

Add the two squared values to find \(c^2\): \(c^2 = 16\). Take the square root of both sides to find \(c\): \(c = \sqrt{16}\).

Get the Value of c

The square root of 16 is 4, so \(c = 4\). Since this is an exact number, no rounding is necessary.

Key concepts

Understanding Right Triangles

Right triangles are a special type of triangle characterized by a 90-degree angle, often called a right angle. This right angle is visually distinguished as a small square at the vertex of the corner. A right triangle consists of three sides: two shorter sides known as the legs, and the longest side called the hypotenuse.

The basic components of a right triangle are:

• The two legs, which form the right angle.
• The hypotenuse, which is opposite the right angle and is always the longest side.

Right triangles are often used in mathematical problems because their properties are well-defined and predictable. Using the Pythagorean theorem, you can explore the relationship between its sides, making it easier to solve many kinds of geometry problems.

Exploring the Hypotenuse

The hypotenuse is a crucial part of the right triangle, representing the side opposite the right angle. It is always the longest side in a right triangle due to the nature of angles and lengths in geometry.

In any right triangle with legs of lengths \(a\) and \(b\), the hypotenuse length \(c\) can be found using the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] This formula asserts that the square of the hypotenuse (\(c^2\)) is equal to the sum of the squares of the other two sides (\(a^2\) and \(b^2\)).

In our example:

• \(a = \sqrt{7}\), thus \(a^2 = 7\)
• \(b = 3\), thus \(b^2 = 9\)
• Therefore, \(c^2 = 7 + 9 = 16\), making \(c = \sqrt{16} = 4\)

Understanding how to find the hypotenuse will allow you to tackle real-life problems involving distances and measurements efficiently.

Calculating with Square Roots

Square roots are fundamental in solving equations involving right triangles, especially when using the Pythagorean theorem. A square root of a number is a value that, when multiplied by itself, gives the original number.

For example, the square root of 16 is 4, because \(4 \times 4 = 16\). In the Pythagorean theorem, solving for a side often involves taking the square root of a sum of squares.

To compute these:

• Identify the squared values: \((\sqrt{7})^2 = 7\) and \(3^2 = 9\)
• Add these to get \(c^2 = 16\)
• Then, to find \(c\), compute \(\sqrt{16} = 4\)

Understanding square roots allows you not merely to compute but also to comprehend the dimensions and relationships within a right triangle.