Please show me how to solve this QuestionSolutionThe center

Please show me how to solve this Question?

Solution

The center of the circle will map out three straight line segments that are of the same length as, and parallel to, the triangle sides, and intersect each other once per corner within the triangle. At each corner of the triangle, the center of the circle will map out an arc that is concave outward, the length of which depends on the angle of the corner. The length of an arc of a circle is the product of the radius of the circle and its central angle in radians. Since we are given r = 1, then these three arc lengths equal the angles in radians.

We are given a triangle of lengths 10, 8, and 6; which has the same proportion as a 3:4:5 right triangle. The angles of such a triangle are 90°, 36.87°, and 53.13°, or 1.571, 0.644, and 0.927 radians. if we draw the situation on paper, we readily see that the angle of the arc drawn out by the center of the circle is related to the angle of the triangle by (arc) = 180° - (angle), or (arc) = - (angle), in radians. Therefore the arc lengths are (remember r = 1): 1.571, 2.498, and 2.214. The total distance is the sum of all these:
1.571 + 2.498 + 2.214 + 10 + 8 + 6 = 30.283.