In ΔPQR, r = 38 cm, mm∠P=49° and mm∠Q=127°. Find the length of p, to the nearest centimeter.

To solve this problem, we will use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

First, we need to find the measure of ∠R. We know that the sum of the angles in a triangle is 180°, so we can calculate ∠R as follows:

∠R = 180° - ∠P - ∠Q ∠R = 180° - 49° - 127° ∠R = 4°

Now we can use the Law of Sines to find the length of side p. We set up the following equation:

p/sin(P) = r/sin(R)

We can rearrange this equation to solve for p:

p = r * sin(P) / sin(R)

Substituting the given values:

p = 38cm * sin(49°) / sin(4°)

Using a calculator to find the sine values and perform the calculation, we find that p is approximately 355 cm.