Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

The correct option is 8.8.

To find the length of the chord $PQ$, we can use a Cartesian coordinate system where the center of the circle $O$ is placed at the origin $(0,0)$.

Step 1: Write the equation of the circle.
Since the circle $C$ has its center at $O(0,0)$ and a radius of $R=5$ meters, its equation is:

$$x^2+y^2=5^2=25$$

Step 2: Determine the coordinates of points A and B.
- Point $A$ is located 4 meters north of $O$, which corresponds to the positive y-axis: $A=(0,4)$.
- Point $B$ is located 3 meters east of $O$, which corresponds to the positive x-axis: $B=(3,0)$.

Step 3: Find the equation of the line passing through A and B.
The line passing through $A(0,4)$ and $B(3,0)$ represents the line containing the chord $PQ$.
Using the intercept form of a linear equation, we get:

$$\frac{x}{3}+\frac{y}{4}=1$$
Multiplying the entire equation by 12 to clear the denominators gives:

$$4x+3y-12=0$$

Step 4: Calculate the perpendicular distance from the center O to the line.
The perpendicular distance $d$ from the origin $(0,0)$ to the line $4x+3y-12=0$ is given by the formula:

$$d=\frac{\left|4\right(0)+3(0)-12|}{\sqrt{4^2+3^2}}$$
Simplifying this expression yields:

$$d=\frac{12}{\sqrt{16+9}}=\frac{12}{\sqrt{25}}=\frac{12}{5}=2.4\text{ meters}$$

Step 5: Determine the length of the chord PQ.
The length of a chord in a circle of radius $R$ at a perpendicular distance $d$ from the center is:

$$\text{Length of PQ}=2\sqrt{R^2-d^2}$$
Substituting $R=5$ and $d=2.4$ into the formula:

$$\text{Length of PQ}=2\sqrt{5^2-{\left(2.4\right)}^2}$$
$$\text{Length of PQ}=2\sqrt{25-5.76}=2\sqrt{19.24}$$
Estimating $\sqrt{19.24}\approx 4.386$:

$$\text{Length of PQ}\approx 2\times 4.386\approx 8.772\text{ meters}$$
Rounding to the nearest option, the length of the chord $PQ$ is nearest to 8.8.