MN and MK are tangent segments drawn to the circle with radius 5 cm. Find MN and MK if MO = 13 cm.

Given:

• Radius of the circle, r = 5 cm
• Distance from the center to the external point M, MO = 13 cm
• MN and MK are tangent segments from point M to the circle.

To find: The lengths of the tangent segments MN and MK.

Solution:

1. Understanding the properties of tangents: A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Therefore, the radius ON is perpendicular to the tangent MN, and the radius OK is perpendicular to the tangent MK. This forms two right-angled triangles, ╨MON╨ and ╨MOK╨, where ╨O╨ is the center of the circle.
2. Applying the Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In ╨MON╨, MO is the hypotenuse, ON is the radius, and MN is the tangent segment.
3. Calculation for MN: Using the Pythagorean theorem: MO^2 = ON^2 + MN^2 Substitute the given values: 13^2 = 5^2 + MN^2 169 = 25 + MN^2 Subtract 25 from both sides: 169 - 25 = MN^2 144 = MN^2 Take the square root of both sides: MN = √144 MN = 12 cm
4. Calculating MK: Since MN and MK are tangent segments drawn from the same external point to the circle, they are equal in length. Therefore, MK = MN.
5. Conclusion: Both tangent segments MN and MK have a length of 12 cm.

Answer: MN = 12 cm, MK = 12 cm