tangent PT, QM = 12, M < P = 30°

Answer 1

First, let's draw a diagram. Draw a circle and a tangent PT to the circle. Draw a point M on the tangent line PT such that length QM = 12. Draw angle PMQ = 30°. Then draw PQ, where Q is the point of tangency.

We want to find length PT. Let's consider angle QPM to be x° .
Since PT is tangent to the circle and PQ is a radius, angle PQM is a right angle (90°). Also, as angle PMQ = 30° , angle QMP = 90° - 30° = 60°.

Now we have a 30-60-90 right triangle - PMQ. For a 30-60-90 triangle, the ratio of the sides is always 1:√3:2. Let's call the side opposite the 30° angle, PQ, "a." Then QM = a√3 and PM = 2a.

We are given that QM = 12, so a√3 = 12. Solving for "a," we find:

a = 12/√3 = 4√3.

Thus, PQ = 4√3 and PM = 2a = 2(4√3) = 8√3.

Finally, we can find PT by adding PQ and QT:

PT = PQ + QT = 4√3 + 12 = 12 + 4√3