<Question>
In $\triangle PQR$, we have $PQ = QR = 34$ and $PR = 32$.  Find the length of median $\overline{QM}$.
</Question>

<think>
<Plan_1>
Okay, so I need to find the length of the median QM in triangle PQR where PQ = QR = 34 and PR = 32. Hmm, let me start by visualizing the triangle. Since PQ and QR are both 34, this is an isosceles triangle with PQ = QR. Wait, but PR is the base with length 32. So the triangle has two equal sides PQ and QR, and the base PR. The median QM is drawn from vertex Q to the midpoint M of side PR. 

First, let me recall what a median is.
In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side. So, since M is the midpoint of PR, QM should split the triangle into two smaller triangles, each with equal area.

Since the triangle is isosceles with PQ = QR, maybe there's some symmetry here that I can exploit. Let me confirm the triangle's structure.

<prompt_1.1> Confirm the structure of the triangle and the position of the median. </prompt_1.1>

Now, to find the length of QM. I remember there's a formula for the length of a median in a triangle, called Apollonius's theorem. Let me recall that.

<prompt_1.2> Recall and verify Apollonius's theorem to find the median. </prompt_1.2>

Alternatively, maybe I can use coordinate geometry. Let me try both methods to confirm.

First, Apollonius's theorem. The formula states that in any triangle, the sum of the squares of any two sides is equal to twice the square of the median to the third side plus twice the square of the half third side. Wait, maybe I should write it down.

<prompt_1.3> Apply Apollonius's theorem to find the median. </prompt_1.3>

Wait, but let me verify this with coordinate geometry to make sure.

Let's place the triangle on a coordinate system.

<prompt_1.4> Verify the result using coordinate geometry by placing the triangle on a coordinate system. </prompt_1.4>

Alternatively, another way to think about it is using the formula for the median in an isosceles triangle. Since the triangle is isosceles with sides PQ = QR = 34, and base PR = 32. The median QM is also the altitude and the angle bisector. Wait, in an isosceles triangle, the median from the apex (vertex Q) to the base (PR) is also the altitude and the angle bisector. So, in this case, QM is not only a median but also an altitude.

<prompt_1.5> Verify the median in an isosceles triangle using the altitude method. </prompt_1.5>
</Plan_1>

<Execution_1>
<execution_1.1>
If PQ and QR are both 34, then the triangle is symmetric with respect to the axis that bisects angle Q and passes through the midpoint of PR. So, point Q is opposite the base PR. Therefore, the median QM is also this axis of symmetry.
</execution_1.1>

<execution_1.2>
Apollollonius's theorem says that in any triangle, the sum of the squares of two sides is equal to twice the square of the median to the third side plus twice the square of the half third side. Wait, the formula is: for triangle ABC, with median AD, then AB² + AC² = 2AD² + 2(BD²). Since BD is half of BC, so the formula becomes AB² + AC² = 2AD² + 2(BC/2)². Simplifying that, it's AB² + AC² = 2AD² + (BC²)/2. Wait, no, let me check again.

Wait, BD is BC/2, so BD² is (BC²)/4. Then, 2*(BD²) is 2*(BC²)/4 = BC²/2. So, the formula is AB² + AC² = 2AD² + BC²/2. Therefore, solving for AD², we get AD² = (2AB² + 2AC² - BC²)/4. Wait, let me verify. Let's rearrange the formula.

If AB² + AC² = 2AD² + (BC²)/2, then 2AD² = AB² + AC² - (BC²)/2. Therefore, AD² = (AB² + AC²)/2 - (BC²)/4.
</execution_1.2>

<execution_1.3>
For triangle PQR, the median QM to side PR can be calculated as:

PQ² + QR² = 2QM² + 2(MR²)

But wait, since M is the midpoint of PR, MR is half of PR. So PR is 32, so MR is 16. Therefore:

34² + 34² = 2QM² + 2*(16)²

Let me compute that step by step.
</execution_1.3>

<execution_1.4>
Let me place points P and R on the x-axis for simplicity. Let's let point P be at (0, 0) and point R be at (32, 0). Then the midpoint M of PR would be at (16, 0). Since the triangle is isosceles with PQ = QR = 34, point Q must lie somewhere along the perpendicular bisector of PR. The perpendicular bisector of PR (which is the x-axis from (0,0) to (32,0)) is the line x = 16. So, point Q has coordinates (16, h), where h is the height of the triangle. 

Now, the distance from Q to P is 34. Let's compute that. The distance between (16, h) and (0,0) is sqrt[(16 - 0)^2 + (h - 0)^2] = sqrt(256 + h^2). This is equal to 34. So:

sqrt(256 + h^2) = 34

Squaring both sides:

256 + h^2 = 1156

Subtract 256:

h^2 = 1156 - 256 = 900

So h = 30 (since height can't be negative, we take the positive root).

Therefore, the coordinates of Q are (16, 30). Now, the median QM is the distance from Q(16,30) to M(16,0). Since both points have the same x-coordinate, the distance is just the difference in y-coordinates: 30 - 0 = 30. So QM = 30.
</execution_1.4>

<execution_1.5>
Wait, is that correct? Wait, if the triangle is isosceles with sides PQ = QR, then the apex is Q, and the base is PR. Therefore, the median from Q to PR is indeed also the altitude and the angle bisector. So, in that case, QM is the altitude, so we can compute its length using the Pythagorean theorem.
</execution_1.5>
</Execution_1>

<Plan_2>
Based on execution_1.3:
34 squared is 34*34. Let me compute that. 30*30 = 900, 30*4 = 120, 4*30=120, 4*4=16. Wait, no, that's not the way. 34 squared is (30 + 4)^2 = 30² + 2*30*4 + 4² = 900 + 240 + 16 = 1156. So 34² = 1156. So both PQ² and QR² are 1156 each. So adding them together: 1156 + 1156 = 2312.

Then, 2*(16)². 16 squared is 256, so 2*256 = 512.

So, plugging into the equation:

2312 = 2QM² + 512

Subtract 512 from both sides: 2312 - 512 = 2QM²

2312 - 512: 2312 - 500 = 1812, then subtract 12 more: 1812 -12 = 1800. So 1800 = 2QM²

Divide both sides by 2: QM² = 900

Then take the square root: QM = sqrt(900) = 30.
-----

Based on execution_1.4, execution_1.5:
Let me check again.

Since QM is the altitude, then in triangle PQM (which is a right triangle), we have PQ² = PM² + QM².

<prompt_2.1> Confirm the median using multiple methods and conclude the result. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Here, PM is half of PR, so 16. PQ is 34. So:

34² = 16² + QM²

Which is again 1156 = 256 + QM²

Subtract 256: QM² = 900, so QM = 30.

So all methods converge to 30.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
**Final Answer**
The length of the median $\overline{QM}$ is \boxed{30}.
</Plan_3>

<Execution_3>
No prompt to execute.
</Execution_3>

<Final_answer>
Therefore, the length of the median $\overline{QM}$ is $\boxed{30}$.
</Final_answer>
</think>