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 that line segment \( AB \) is a chord of a circle with radius \( R = 5 \) meters, and the coordinates of points \( A \) and \( B \) form a right triangle with legs of 3 m and 4 m respectively, we are to find the length of the chord \( PQ \), where \( PQ \) is symmetric about center \( O \).

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

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

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

Using the formula for perpendicular from center to chord: \[ OT = \frac{\text{Product of legs}}{AB} = \frac{3 \times 4}{5} = \frac{12}{5} = 2.4 \text{ m} \]

Step 3: Use Pythagoras Theorem in triangle \( \triangle OTQ \)

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

Step 4: Use symmetry to find full chord \( PQ \)

Since \( OT \) is the perpendicular from center, it bisects the chord: \[ PQ = 2 \times TQ = 2 \times 4.4 = 8.8 \text{ m} \]

Final Answer:

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

Answer 2

Given:

• \( OA = 4 \, \text{m} \)
• \( OB = 3 \, \text{m} \)
• \( AB = 5 \, \text{m} \)

 

Step 1: Let OR be the perpendicular from center O bisecting chord AB

Let \( a = QC \), \( b = PC \), then: \[ a + b = AB = 5 \tag{1} \] Since OR is perpendicular, apply Pythagoras: \[ 4^2 - k^2 = a^2 \tag{2} \] \[ 3^2 - k^2 = b^2 \tag{3} \] Subtracting (3) from (2): \[ a^2 - b^2 = (4^2 - k^2) - (3^2 - k^2) = 16 - 9 = 7 \tag{4} \]

Step 2: Solve using equations (1) and (4)

From equation (4), and knowing: \[ a^2 - b^2 = (a + b)(a - b) = 5(a - b) = 7 \Rightarrow a - b = \frac{7}{5} = 1.4 \tag{5} \] Solving equations (1) and (5): 
Add: \( a + b = 5 \), \( a - b = 1.4 \) 
\[ \Rightarrow 2a = 6.4 \Rightarrow a = 3.2, \quad b = 1.8 \]

Step 3: Compute \( k^2 \)

Using equation (2): \[ k^2 = 4^2 - a^2 = 16 - (3.2)^2 = 16 - 10.24 = 5.76 \]

Step 4: Use triangle \( \triangle POC \) and \( \triangle QOC \)

From \( \triangle POC \): \[ OP^2 = (x + a)^2 + k^2 = 25 \Rightarrow (x + 3.2)^2 = 25 - 5.76 = 19.24 \Rightarrow x + 3.2 = \sqrt{19.24} \approx 4.39 \Rightarrow x \approx 1.19 \, \text{m} \] Similarly, from \( \triangle QOC \): \[ (y + b)^2 = 25 - 5.76 = 19.24 \Rightarrow y + 1.8 = \sqrt{19.24} \approx 4.39 \Rightarrow y \approx 2.59 \, \text{m} \]

Step 5: Compute Total Length of \( PQ \)

\[ PQ = PC + AB + QC = x + AB + y = 1.19 + 5 + 2.59 = 8.78 \approx \boxed{8.8 \, \text{m}} \]

Final Answer:

Option (D): \( \boxed{8.8 \, \text{meters}} \)