To solve this, we will use the Intersecting Chords Theorem (also known as the Power of a Point theorem). This theorem states that if two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

In this case, the chords $MN$ and $OP$ intersect at point $Q$.

According to the theorem:

$MQ \cdot QN = OQ \cdot QP$

We are given:

• $MN = 32$
• $OQ = 23$
• $QP = 20$

Let:

• $MQ = x$
• $QN = 32 - x$

Now, applying the theorem:

$x \cdot (32 - x) = 23 \cdot 20$

This simplifies to:

$x \cdot (32 - x) = 460$

Expanding:

$32x - x^2 = 460$

Rearranging into a quadratic equation:

$x^2 - 32x + 460 = 0$

We will now solve this quadratic equation to find $x = MQ$.It seems the discriminant is negative, which indicates no real solutions exist for this quadratic equation. This suggests that there might have been an issue in setting up the problem.

Let me double-check the values and calculations to ensure the setup is correct. I'll recompute the discriminant and explain if adjustments are necessary.The discriminant is $-816$, which is negative, confirming that no real solutions exist for the quadratic equation as currently set up. This suggests that the provided measurements may have an inconsistency.

Could you confirm the given values for the problem, or would you like assistance in further investigating the setup?