K, L, and M are three points on a circle of radius 8 cm such that KLM is an equilateral triangle. What is the shortest distance of any side of the triangle from the center of the circle?

Question

15-09-2023
Mathematics

College

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K, L, and M are three points on a circle of radius 8 cm such that KLM is an equilateral triangle. What is the shortest distance of any side of the triangle from the center of the circle?

Answer :

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TaylorSmith4590 TaylorSmith4590 · Answer 1 · 2023-12-28T00:02:46-05:00

Final answer:

To find the shortest distance from the centre of a circle to a side of an inscribed equilateral triangle, one must draw a perpendicular from the centre to the side, forming a 30-60-90 triangle, and then use the known radius of the circle to find that distance. Therefore, the shortest distance from the centre of the circle to any side of the triangle KLM is (8/√3) cm.

Explanation:

The student has asked about the shortest distance from the centre of the circle to any side of an equilateral triangle inscribed within it. Given that the circle has a radius of 8 cm, we can determine this distance using the properties of equilateral triangles and circles.

Firstly, let's draw the triangle KLM inscribed in the circle. Since KLM is equilateral, each angle is 60°. If we draw a perpendicular line from the centre of the circle to any side of the triangle, this line will bisect that side and form two 30-60-90 right triangles. Let's call the midpoint of any side of the triangle 'N', and the distance from the centre 'O' to the midpoint 'N' is what we are looking for.

In a 30-60-90 triangle, the ratio of the lengths of the sides opposite the 30°, 60°, and 90° angles is 1:√3:2. Since the hypotenuse 'OK' (radius of the circle) is given as 8 cm, the line 'ON', which is opposite the 60° angle, is (8/√3) cm. Therefore, the shortest distance from the centre of the circle to any side of the triangle KLM is (8/√3) cm.