A triangle has one side that is 4 inches long and another side that is 4.5 inches long. The angle that is between them is 80°.

and?

To find the length of the third side of the triangle, you can use the Law of Cosines. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides multiplied by the cosine of the included angle.

The formula for the Law of Cosines is:

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

Where:
- c is the length of the side opposite the angle we want to find (in this case, the third side)
- a and b are the lengths of the other two sides
- C is the included angle (in this case, 80°)

Let's plug in the values into the formula and solve for c:

c^2 = (4 inches)^2 + (4.5 inches)^2 - 2 * 4 inches * 4.5 inches * cos(80°)

Simplifying:

c^2 = 16 inches^2 + 20.25 inches^2 - 36 inches^2 * cos(80°)

c^2 = 16 + 20.25 - 36 * cos(80°)

c^2 = 36.25 - 36 * cos(80°)

Now, calculate c:

c = sqrt(36.25 - 36 * cos(80°))

Using a calculator:

c ≈ sqrt(1.921) ≈ 1.385 inches

Therefore, the length of the third side of the triangle is approximately 1.385 inches.