The method for finding the missing side of a triangle depends on what information you are given about the triangle. Here are three common scenarios:
1. Right Triangle (You know two sides):
If you have a right triangle (contains a 90degree angle) and know the lengths of two sides, you can use the Pythagorean Theorem to find the missing side.

The Pythagorean Theorem states: a² + b² = c²

Where:

a and b are the lengths of the two known sides (legs) of the right triangle. It doesn't matter which is a and which is b.

c is the length of the missing side (hypotenuse), which is always the side opposite the right angle.

Steps:

Square the values of the two known side lengths (a² and b²).

Add the squares together (a² + b²).

Take the square root of the sum (√(a² + b²)).

This will give you the length of the missing side (hypotenuse).
2. Triangle (You know one side and two angles):
If you know the length of one side (let's call it base B) and two angles (let's call them angles A and C), you can use the Law of Sines to find the missing sides.

The Law of Sines states: a / sin(A) = b / sin(B) = c / sin(C)

Where:

a, b, and c represent the lengths of the sides of the triangle.

A, B, and C represent the measure of the corresponding opposite angles.

B is the known side length (base).

You are trying to find the missing side length (either a or c depending on which angle you know besides B).

Steps:

Identify the missing side you want to find (a or c).

Look at the ratio with the missing side length (a / sin(A) or c / sin(C)). Make sure the angle in the ratio (A or C) is one of the two known angles.

Set up a proportion with the known side length (B) and its opposite angle (B). B / sin(B) = (known value).

Crossmultiply and solve for the missing side length (a or c).
3. Triangle (You know two side lengths and the angle between them):
If you know the lengths of two sides (let's call them side A and side B) and the angle between them (let's call it angle C), you can use the Law of Cosines to find the missing side (opposite to angle C).

The Law of Cosines states: c² = a² + b²  2ab * cos(C)

Where:

a and b are the lengths of the two known sides (A and B).

c is the length of the missing side (opposite angle C).

C is the angle measure between the two known sides (A and B).

Steps:

Square the values of the two known side lengths (a² and b²).

Multiply the product of the two known side lengths by the cosine of the angle between them (2ab * cos(C)).

Add the squares of the two known sides (a² + b²) and subtract the product from step 2.

Take the square root of the result (√(a² + b²  2ab * cos(C))).

This will give you the length of the missing side (opposite angle C).
By understanding these methods and the information you are given about the triangle, you can effectively find the missing side length.