Question

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

Accepted Answer

C. Set $O$ as the origin $(0,0)$. $A = (0, 4)$ and $B = (3, 0)$. The line passing through $A$ and $B$ has equation $\frac{x}{3} + \frac{y}{4} = 1 \implies 4x + 3y - 12 = 0$. The perpendicular distance $d$ from $O(0,0)$ to line $AB$ is: $$d = \frac{|12|}{\sqrt{4^2 + 3^2}} = \frac{12}{5} = 2.4	ext{ meters}$$ The length of chord $PQ$ in a circle of radius $R = 5$ with distance $d = 2.4$ from the center is: $$PQ = 2\sqrt{R^2 - d^2} = 2\sqrt{5^2 - 2.4^2} = 2\sqrt{25 - 5.76} = 2\sqrt{19.24} \approx 2 	imes 4.386 = 8.77	ext{ m} \approx 8.8	ext{ m}$$

Length of Chord in Circle

Explanation

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