Answer:
Explanation:
Let O be the center of the circle, and let x be the length of the radius of the circle. Since segment DE is a minor arc, it subtends an angle of less than 180 degrees at the center of the circle, and its length is given by:
length of DE = (angle AOB / 360) * 2 * pi * x
We are given that the length of DE is 52 cm, and we know that angle AOB is 2 times angle ACB, since these angles subtend the same arc. Therefore, we have:
angle AOB = 2 * angle ACB
We can use the law of cosines to find angle ACB:
cos(ACB) = (AB^2 + BC^2 - AC^2)/(2 * AB * BC)
cos(ACB) = (25 + 64 - 81)/(2 * 5 * 8)
cos(ACB) = -1/8
Since angle ACB is acute, we have:
ACB = arccos(-1/8)
ACB = 100.14 degrees (rounded to two decimal places)
Therefore, angle AOB is twice this angle, or:
angle AOB = 200.28 degrees
Substituting the values of (angle AOB, x) into the equation for the length of DE, we get:
52 = (200.28 / 360) * 2 * pi * x
Simplifying and solving for x, we get:
x = 26 / pi
The circumference of the circle is given by:
circumference = 2 * pi * x
Substituting the value of x, we get:
circumference = 2 * pi * (26 / pi) = 52 cm
Therefore, the circumference of circle F is 52 cm.