Given two concentric circles, with radii 15 cm and 9 cm, what is the length of the chord MN of the larger circle that is tangent to the smaller circle?

To find the length of the chord MN of the larger circle that is tangent to the smaller circle, we can use the properties of right triangles and the Pythagorean theorem.

First, identify the given values:

• The radius of the larger circle (R) is 15 cm.

• The radius of the smaller circle (r) is 9 cm.

Since the chord MN is tangent to the smaller circle, we know the distance from the center of the circles to the tangential point (which lies on the smaller circle) is equal to the radius of the smaller circle. This creates a right triangle where:

• One leg is the radius of the smaller circle (9 cm).

• The other leg is the segment from the center of the larger circle to the midpoint of the chord MN.

The distance from the center of the larger circle to the midpoint of the chord is equal to the radius of the larger circle (15 cm) minus the radius of the smaller circle (9 cm), which gives us:

Distance from center to midpoint of MN = 15 cm - 9 cm = 6 cm.

Now we can apply the Pythagorean theorem in this right triangle, where the hypotenuse is the radius of the larger circle