Question

If 3 and 4 are the legs of a right triangle, the hypotenuse is ____.

Short answer

The hypotenuse is 5.

Step-by-step solution

- Understand the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). Mathematically, it can be written as: \[ c^2 = a^2 + b^2 \] where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs.

- Identify the lengths of the legs

In this problem, the lengths of the legs are given as 3 and 4. Thus, \(a = 3\) and \(b = 4\).

- Plug the lengths into the Pythagorean theorem

Replace \(a\) and \(b\) in the Pythagorean theorem with the given values: \[ c^2 = 3^2 + 4^2 \]

- Calculate the squares of the legs

Compute the squares: \[ 3^2 = 9 \] \[ 4^2 = 16 \]

- Add the squares

Add the results from the previous step: \[ 9 + 16 = 25 \]

- Solve for the hypotenuse

To find the hypotenuse \(c\), take the square root of 25: \[ c = \sqrt{25} = 5 \]

Key concepts

right-angled triangle

To grasp how to find the hypotenuse using the Pythagorean theorem, we first need to understand what a right-angled triangle is. A right-angled triangle, also known as a right triangle, is a type of triangle that includes one angle that is exactly 90 degrees. This 90-degree angle is called the right angle. The side directly opposite this right angle is known as the hypotenuse, and it is always the longest side of the triangle.

The other two sides are referred to as the legs of the triangle. These legs form the right angle. In real-world terms, think of the right angle being the corner of a piece of paper, with the hypotenuse being a diagonal cut across the paper. This setup is fundamental in many mathematical and geometrical applications, including construction and navigation. In our exercise, the given right-angled triangle has legs of length 3 and 4.

hypotenuse calculation

Once you know what a right-angled triangle is, the next step is to learn how to calculate the hypotenuse. This is where the Pythagorean theorem comes into play. The Pythagorean theorem states that in any right-angled triangle, the square of the length of the hypotenuse (denoted as \(c\)) is equal to the sum of the squares of the lengths of the other two sides (the legs, denoted as \(a\) and \(b\)). In formula terms, this is written as:
\[ c^2 = a^2 + b^2 \]
Using the given problem, we know the lengths of the legs:\( a = 3 \) and \( b = 4 \). We substitute these values into the formula to find \( c \):
\[ c^2 = 3^2 + 4^2 \]
Calculating this gives us:
\[ c^2 = 9 + 16 \]
\[ c^2 = 25 \]
To find \( c \), which is the hypotenuse, we then take the square root of 25:
\[ c = \sqrt{25} = 5 \]
This means the hypotenuse of our right-angled triangle is 5.

squares of the legs

An essential part of using the Pythagorean theorem is understanding what the 'squares of the legs' are. The 'legs' of the triangle are the other two sides that make up the right angle. For our example, the legs measure 3 and 4 units. When we talk about 'squaring' these lengths, we mean multiplying each leg by itself. In mathematical language, squaring transforms a number \(x\) into \(x^2\).
Using the legs from our triangle:
\[ 3^2 = 3 * 3 = 9 \]
\[ 4^2 = 4 * 4 = 16 \]
This gives us the 'squared values' of the legs. According to the Pythagorean theorem, the sum of these squared values will be equal to the square of the hypotenuse. In this exercise, adding \( 9 \) and \( 16 \) together yields:
\[ 9 + 16 = 25 \]
This sum (25) tells us how the lengths of the legs relate to the hypotenuse. To find the actual length of the hypotenuse, we take the square root of the sum, resulting in a hypotenuse length of 5.