Okay, so I need to find the length of the median QM in triangle PQR where PQ and QR are both 34 units long, and PR is 32 units. Let me start by visualizing the triangle. Since PQ equals QR, triangle PQR is isosceles with Q as the apex. That means the base is PR, which is 32 units long. The median QM should connect vertex Q to the midpoint M of side PR. 

First, I remember that in an isosceles triangle, the median from the apex also acts as the altitude and the angle bisector. Wait, but here we're dealing with the median from Q to PR. Since PR is the base, and Q is the apex, the median QM should split PR into two equal parts. So, the midpoint M of PR will be 16 units from both P and R because PR is 32. That makes sense.

Now, to find the length of QM, I can use the formula for the length of a median in a triangle. The formula is:

\[
\text{Median length} = \sqrt{\frac{2a^2 + 2b^2 - c^2}{4}}
\]

Where a and b are the lengths of the sides adjacent to the median, and c is the length of the side to which the median is drawn. Wait, let me confirm that. Alternatively, since this is an isosceles triangle, maybe there's a simpler approach. Let me think.

Since triangle PQR is isosceles with PQ = QR = 34 and PR = 32, the median QM is also the altitude. So, if I can compute the height of the triangle from Q to PR, that should be the length of QM. 

To find the height, I can split the isosceles triangle into two right triangles by drawing the median QM. Each right triangle will have half of PR as one leg (16 units), the height QM as the other leg, and PQ as the hypotenuse (34 units). Then, using the Pythagorean theorem:

\[
QM^2 + 16^2 = 34^2
\]

Let me calculate that. First, compute 34 squared: 34*34. Let's see, 30 squared is 900, and 4 squared is 16, and the cross term is 2*30*4 = 240. So, 900 + 240 + 16 = 1156. So, 34^2 is 1156. Then, 16 squared is 256. So,

\[
QM^2 = 1156 - 256 = 900
\]

Therefore, QM is the square root of 900, which is 30. So, the length of the median QM is 30 units. 

Wait, let me double-check that. If the triangle is isosceles with sides 34, 34, and base 32, then splitting it into two right triangles with legs 16 and QM, and hypotenuse 34. Then, 16^2 + QM^2 = 34^2. 256 + QM^2 = 1156. So, QM^2 = 900, so QM = 30. That seems correct. 

Alternatively, using the median formula. Let me try that method just to confirm. The formula for the median from Q to side PR is:

\[
QM = \sqrt{\frac{2PQ^2 + 2QR^2 - PR^2}{4}}
\]

But since PQ = QR = 34 and PR = 32, substituting:

\[
QM = \sqrt{\frac{2(34)^2 + 2(34)^2 - (32)^2}{4}} = \sqrt{\frac{2*1156 + 2*1156 - 1024}{4}} = \sqrt{\frac{2312 + 2312 - 1024}{4}} = \sqrt{\frac{4624 - 1024}{4}} = \sqrt{\frac{3600}{4}} = \sqrt{900} = 30
\]

Same result. So that's consistent. Therefore, the length of the median QM is 30. 

Alternatively, using coordinate geometry. Let me place the triangle on a coordinate plane to verify. Let’s place point P at (-16, 0) and point R at (16, 0), so that PR is 32 units long. Then, since Q is the apex of the isosceles triangle, it should be somewhere along the y-axis. Let's call Q's coordinates (0, h). Then, the distance from Q to P is 34. 

The distance from Q(0, h) to P(-16, 0) is sqrt[(-16 - 0)^2 + (0 - h)^2] = sqrt[256 + h^2] = 34. Therefore:

\[
256 + h^2 = 34^2 = 1156
\]
\[
h^2 = 1156 - 256 = 900
\]
\[
h = 30
\]

Therefore, the coordinates of Q are (0, 30). The midpoint M of PR is at (0, 0), since PR is from (-16, 0) to (16, 0). So, the distance from Q(0,30) to M(0,0) is sqrt[(0 - 0)^2 + (0 - 30)^2] = sqrt[900] = 30. That confirms it again. 

So all three methods give the same answer. I think that's solid. The key steps were recognizing the isosceles triangle, realizing the median is also the altitude, applying the Pythagorean theorem, and verifying with the median formula and coordinate geometry. All methods lead to the median length being 30. 

I don't see any mistakes in the calculations. Each step checks out, and the different approaches confirm the same result. Therefore, I'm confident the answer is 30.
Therefore, the length of the median $\overline{QM}$ is $\boxed{30}$.
</think>