# How do you find the third side of a triangle?

#### Understand the Problem

The question is asking for a method to determine the length of the third side of a triangle when the lengths of the other two sides are known. This typically involves applying geometric principles such as the Pythagorean theorem if the triangle is a right triangle, or using the Law of Cosines or Law of Sines for other types of triangles.

Use either the Pythagorean theorem or the Law of Cosines.

The final answer depends on the triangle type. For a right triangle, use the Pythagorean theorem. For other triangles, use the Law of Cosines.

#### Steps to Solve

1. Identify the type of triangle

Determine whether the triangle is a right triangle, an acute triangle, or an obtuse triangle. This affects which formula you'll use.

1. Use the Pythagorean Theorem for right triangles

If the triangle is a right triangle, use the Pythagorean theorem:

$$c^2 = a^2 + b^2$$

Where $c$ is the hypotenuse and $a$ and $b$ are the other two sides. Solve for the unknown side.

1. Use the Law of Cosines for non-right triangles

For other types of triangles, use the Law of Cosines:

$$c^2 = a^2 + b^2 - 2ab \cos(C)$$

Where $c$ is the side you want to find, $a$ and $b$ are the known sides, and $C$ is the angle opposite side $c$. Solve for $c$.

The final answer depends on the triangle type. For a right triangle, use the Pythagorean theorem. For other triangles, use the Law of Cosines.

Knowing which formula to use depends on understanding the properties of the triangle. The Pythagorean theorem is simpler but only applies to right triangles, while the Law of Cosines is more versatile and can be used for any triangle.

#### Tips

Common mistakes include using the Pythagorean theorem for non-right triangles and forgetting to use proper angle measures (in radians or degrees) for the Law of Cosines.

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