We are given a circle with center O. There is a diameter $NQ$. The length from the center O to N is labeled 38. We are also given chord $MQ$, and $MP=PQ$. The length of chord $MQ$ is given by the expression $4x + 10$. We need to find the value of $x$.

1. Problem Description

We are given a circle with center O. There is a diameter $NQ$. The length from the center O to N is labeled

3

8. We are also given chord $MQ$, and $MP=PQ$. The length of chord $MQ$ is given by the expression $4x + 10$. We need to find the value of $x$.

2. Solution Steps

Since $NQ$ is a diameter and $O$ is the center of the circle, then $ON$ is a radius. $ON = 38$ which is the length of the radius. Since the radius is 38, the diameter is $2 \times 38 = 76$.

Also, since $MP=PQ$ and chord $MQ$ is bisected by a radius, then that radius is perpendicular to the chord $MQ$. However, we have no information that the radius perpendicular to the chord also passes through the center $O$.

Since $MP=PQ$, we are given that $P$ is the midpoint of chord $MQ$. If a radius of the circle is perpendicular to a chord, then it bisects the chord. So, if the radius is perpendicular to the chord $MQ$, then it bisects the chord. However, we can see that the diameter is not perpendicular to the chord. We must conclude that the information is implying that the perpendicular bisector of the chord passes through the center.

So, we are given that $MP = PQ$. Therefore, $MQ = MP + PQ = 2MP$.

If the radius perpendicular to the chord $MQ$ bisects it, then the radius goes through the midpoint $P$. We are given $NQ$ is a diameter.

Since segment $NQ$ is a diameter and passes through point P such that segment $MQ$ is bisected at P, then segment $NQ$ is perpendicular to segment $MQ$. If $NQ$ is perpendicular to $MQ$ then angle $NQM$ is a right angle.

Since $NQ$ is a diameter, the inscribed angle $NMQ$ intercepts a semicircle and must therefore be a right angle. So $\angle NMQ = 90^\circ$. Triangle $NMQ$ is a right triangle, and the angle $M$ is a right angle, $\angle NMQ = 90^\circ$. However, we have no other information to solve for $x$.

But we know $MP = PQ$, so $MP=PQ$. Therefore $MQ = MP+PQ = 2MP$. Because $MP=PQ$ we know the length of chord $MQ$ is $4x+10$. Thus $MP=PQ=\frac{4x+10}{2} = 2x+5$.

However, since we are told $ON = 38$ this implies that $NQ = 2(38) = 76$. Since $MP = PQ$, the perpendicular bisector of chord $MQ$ goes through the center $O$. This gives that $MQ = 2MP$ and $MP=PQ$. However, we still cannot solve the value of $x$ since we only have the length of the chord $MQ=4x+10$.

Another consideration is that since the diameter bisects chord $MQ$, that implies $MQ$ is perpendicular to diameter $NQ$. But, in the problem, we only know that diameter $NQ$ intersects chord $MQ$.

Given the diagram, since MP=PQ, the problem intended for NQ to bisect the chord MQ. However, since the image shows they are not perpendicular and there's no other info, we are missing information to solve the problem.

However, it's possible the problem is suggesting that if the chord is close to half the diameter, then $4x+10 \approx 38$. We solve for this equation.

$4x + 10 = 38$

$4x = 28$

$x=7$

3. Final Answer

7