Find the length of the third side of the triangle. If necessary, write in simplest radical form.
Understand the Problem
The question asks us to find the missing side of the triangle. We can use the Pythagorean theorem $a^2 + b^2 = c^2$ to find the length of the missing side. In this case we can say that $1^2 + b^2 = (\sqrt{2})^2$, solve for b, the missing side.
Answer
$1$
Answer for screen readers
$1$
Steps to Solve
- Identify the Pythagorean Theorem
The Pythagorean Theorem states that in a right 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 lengths of the other two sides. We can write this as:
$$a^2 + b^2 = c^2$$
where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides (legs).
- Plug in the given values
We are given that one leg, $a$, has a length of 1, and the hypotenuse, $c$, has a length of $\sqrt{2}$. We need to find the length of the other leg, $b$. Plug the values into the Pythagorean Theorem:
$$1^2 + b^2 = (\sqrt{2})^2$$
- Simplify the equation
Simplify the squares:
$$1 + b^2 = 2$$
- Solve for $b^2$
Subtract 1 from both sides of the equation:
$$b^2 = 2 - 1$$
$$b^2 = 1$$
- Solve for $b$
Take the square root of both sides of the equation:
$$b = \sqrt{1}$$
$$b = 1$$
$1$
More Information
The third side of the triangle is equal to 1.
Tips
A common mistake when using the Pythagorean theorem is to incorrectly identify the hypotenuse and the legs of the right triangle. The hypotenuse is always the side opposite the right angle, and it's the longest side. Also, forgetting to take the square root at the end to find the side length is a frequent error.
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