What is the length of the side M in triangle MNP given M = 35 degrees and P = 65 degrees and p = 20?'

This is a trigonometry question and used "The Law of Sines."

Perhaps you have learned that in a right triangle, the sine equals the opposite side over the hypotenuse.

The Law of Sines is different. It applies to ALL triangles, not just right triangles.

Basically, it says that the ratio of the sine of any angle to the opposite side is equal to the same ratio with either of the other angles.

The three-part formula looks like this:

sinA / a = sinB / b = sinC / c

Using two of the ratios from the formula, if you know three parts of the triangle, you can find the fourth part.

So in your triangle,

sin P / p = sin M / m

Plugging in what you know ...

sin 65 / 20 = sin 35 / m

Now solve for m by cross multiplying ...

m (sin 65) = 20 (sin 35)

Then

m = 20 (sin 35) / sin 65

Now go to the calculator ...

m = 12.657