Question

A chord of a circle is 12 cm in length, and its distance from the center is 8 cm. Find the length of the chord of the same circle which is at a distance of 6 cm from the center?

a) 10 cm
b) 14 cm
c) 16 cm
d) 18 cm

Answer 1

Using the Pythagorean theorem, the length of the second chord at a distance of 6 cm from the center of the circle is determined to be 16 cm, making option C the correct answer.

Step-by-step explanation:

To find the length of the chord which is 6 cm away from the center of the circle, we can use the relationship between the chord length, the radius of the circle, and the perpendicular distance from the center to the chord. Consider that we have two chords, one with a length 12 cm and a distance 8 cm from the center, and another one whose length we are to determine, with a distance of 6 cm from the center.

Step-By-Step Explanation:

1. Let's denote the radius of the circle as r.
2. The original chord AB is 12 cm creates a right triangle with the radius and the perpendicular distance from the center to the chord.
3. Using the Pythagorean theorem, we can find the radius r: r^2 = 8^2 + (12/2)^2, which gives r = 10 cm.
4. Now, we look for the second chord CD, which is perpendicular to another radius and is 6 cm from the center.
5. Again, applying the Pythagorean theorem, the half-length of CD is found by sqrt(r^2 - 6^2), which equals sqrt(10^2 - 6^2) = sqrt(64) = 8 cm.
6. The full length of chord CD is therefore 2 × 8 cm = 16 cm.

This calculation confirms that Option C, 16 cm, is the correct answer.