Question 1

The perpendicular from the center of a circle to a chord bisects the chord.

Step 1 — Note the given values

Let's identify what we know. The radius of the circle is 7 cm. The perpendicular distance from the center to the chord is 6 cm.

Step 2 — Find half the chord length

We can form a right-angled triangle. The radius is the hypotenuse. The perpendicular distance is one leg. Half the chord is the other leg. Let the half-chord be AM. We use the Pythagorean theorem.

$OA^2 = OM^2 + AM^2$

$7^2 = 6^2 + AM^2$

$49 = 36 + AM^2$

$AM^2 = 49 - 36$

$AM^2 = 13$

$\boxed{AM = \sqrt{13} \text{ cm}}$

Step 3 — Calculate the full chord length

The perpendicular bisects the chord. So, the full chord length is twice AM. Let the chord be AB.

$AB = 2 \times AM$

$AB = 2 \times \sqrt{13}$

$\boxed{AB = 2\sqrt{13} \text{ cm}}$

Answer

The length of the chord is $2\sqrt{13} \text{ cm}$.