Question

If two concentric circles of radii 5 cm and 3 cm are drawn, then the length of the chord of the larger circle which touches the smaller circle is:

• A. 4 cm
• B. 6 cm
• C. 8 cm
• D. 10 cm

Hint:

Use the Pythagorean theorem to find the length of the chord when the distance from the center to the chord and the radius of the circle are known.

Answer 1

The Correct Option is B

The length of the chord of the larger circle that touches the smaller circle can be found using the Pythagorean theorem. Let the distance from the center of the circles to the chord be the radius of the smaller circle, \(r = 3 \, \text{cm}\). The radius of the larger circle is \(R = 5 \, \text{cm}\). The length of the chord is the distance across the larger circle, and the perpendicular distance from the center to the chord is \(3 \, \text{cm}\). Thus, we can use the Pythagorean theorem to find the length of the half chord: \[ \left(\frac{l}{2}\right)^2 + 3^2 = 5^2 \] \[ \left(\frac{l}{2}\right)^2 + 9 = 25 \quad \Rightarrow \quad \left(\frac{l}{2}\right)^2 = 16 \quad \Rightarrow \quad \frac{l}{2} = 4 \] Thus, the length of the chord is \(l = 8\) cm. Therefore, the correct answer is option (2).