Question

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

• A. 7.2
• B. 7.8
• C. 6.6
• D. 8.8

Answer 1

The Correct Option is D

The correct answer is (D): \(8.8\)

Given a circle with center O and radius \( R = 5 \) meters, and a chord \( AB \) whose endpoints form a right triangle with legs of 3 m and 4 m, determine the length of chord \( PQ \), which is symmetric about the center O.

Step 1: Calculate the length of chord \( AB \)

\[ AB = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 2: Compute the perpendicular distance \( OT \) from the center O to chord \( AB \)

The perpendicular distance from the center to a chord is calculated as: \[ OT = \frac{\text{Product of legs}}{AB} = \frac{3 \times 4}{5} = \frac{12}{5} = 2.4 \text{ m} \]

Step 3: Apply the Pythagorean Theorem to triangle \( \triangle OTQ \)

With radius \( OQ = 5 \) and \( OT = 2.4 \), we find \( TQ \): \[ TQ = \sqrt{OQ^2 - OT^2} = \sqrt{5^2 - (2.4)^2} = \sqrt{25 - 5.76} = \sqrt{19.24} \approx 4.4 \text{ m} \]

Step 4: Determine the full chord length \( PQ \) using symmetry

The perpendicular from the center bisects the chord, so: \[ PQ = 2 \times TQ = 2 \times 4.4 = 8.8 \text{ m} \]

Final Answer:

\[ \boxed{PQ = 8.8 \text{ meters}} \]