Question

In the figure \(OB = 13\) cm, \(OP⊥AB\), \(OP=12\) cm then \(AB =\)

• A. 100 cm
• B. 50 cm
• C. 75 cm
• D. 10 cm

Answer 1

The Correct Option is D

To solve the problem, we need to find the length of \( AB \) given that \( OB = 13 \, \text{cm} \), \( OP = 12 \, \text{cm} \), and \( OP \perp AB \). Here's the step-by-step solution:

Step 1: Understand the geometry.
The point \( O \) is the center of the circle, and \( OB \) is the radius of the circle. Since \( OP \perp AB \), \( P \) is the midpoint of \( AB \). This means \( AP = PB \), so \( AB = 2 \times AP \).

Step 2: Use the Pythagorean theorem in \( \triangle OPA \).
In \( \triangle OPA \), \( OA \) is the hypotenuse (which is the radius of the circle, \( OB = 13 \, \text{cm} \)), \( OP = 12 \, \text{cm} \), and \( AP \) is the unknown side. By the Pythagorean theorem:

\[ OA^2 = OP^2 + AP^2 \]

Substituting the known values:

\[ 13^2 = 12^2 + AP^2 \]

Simplify:

\[ 169 = 144 + AP^2 \]

Solve for \( AP^2 \):

\[ AP^2 = 169 - 144 = 25 \]

Take the square root of both sides:

\[ AP = \sqrt{25} = 5 \, \text{cm} \]

Step 3: Find the length of \( AB \).
Since \( P \) is the midpoint of \( AB \), we have:

\[ AB = 2 \times AP = 2 \times 5 = 10 \, \text{cm} \]

Final Answer:
The length of \( AB \) is \( 10 \, \text{cm} \).

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