Problem 95 - Elementary Algebra

Question

If the two legs of a right triangle both measure 1 unit, then find the length of the hypotenuse.

Step-by-Step Solution

Answer

The hypotenuse is \( \sqrt{2} \) units long.

Step 1: Understand the Problem

We are given a right triangle with both legs measuring 1 unit. We need to find the length of the hypotenuse.

Step 2: Apply the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.

Step 3: Substitute the Known Values

Here, \( a = 1 \) and \( b = 1 \). Substitute these into the theorem: \( 1^2 + 1^2 = c^2 \).

Step 4: Calculate the Squares

Calculate \( 1^2 \) and \( 1^2 \). We get: \( 1 + 1 = c^2 \).

Step 5: Solve for \( c \)

Simplify the left side to get \( 2 = c^2 \).

Step 6: Take the Square Root

Take the square root of both sides to find \( c \). We get \( c = \sqrt{2} \).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one of its angles exactly equal to 90 degrees. This right angle is a defining feature. In a right triangle, the sides have specific roles:

• Two sides form the right angle, these are known as the "legs".
• The longest side, which is opposite the right angle, is known as the "hypotenuse".

The right triangle is a key element in trigonometry due to its fixed relationships between angles and sides. This makes it useful in solving many types of mathematical problems, including finding distances and angles.

Hypotenuse

The hypotenuse is always opposite the right angle in a right triangle. It is the longest side of the triangle. This is important because the Pythagorean theorem relies on understanding and identifying the hypotenuse.

In practice, to find the hypotenuse we use the Pythagorean theorem:

• The formula is \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the two legs and \( c \) is the hypotenuse.

To solve for the hypotenuse when the lengths of both legs are known, you need to compute the sum of the squares of the legs and then find the square root of that sum. In the given exercise example, where both legs are 1 unit long, the hypotenuse calculated using the formula is \( \sqrt{2} \). This value is derived by first squaring the legs, adding them, and then taking the square root.

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For instance, the square root of 4 is 2, since \( 2 \times 2 = 4 \).

The concept of square root is crucial in the Pythagorean theorem. After finding that the square of the hypotenuse \( c^2 = 2 \) in the exercise, you take the square root to solve for \( c \).

• This calculation simplifies being able to express the length of the hypotenuse as \( \sqrt{2} \).
• Understanding how to operate with square roots enhances your ability to solve exercises involving right triangles and the Pythagorean theorem efficiently.

By mastering the use of square roots, students can easily transition between squared numbers and their roots, making problem-solving smoother.