The polygon circumscribes a circle that has 4 sides, the top having 7 cm and 16cm and the bottom having 4 cm and 6 cm. What is the perimeter of the polygon?
Let's draw the polygon and the circle it circumscribes:
We know that the circle intersects the midpoint of each side of the polygon, and that the distance from the center of the circle to each vertex of the polygon is equal to the radius of the circle.
Let's call the radius of the circle "r". We can see that the height of the trapezoid (the distance between the parallel sides) is equal to the diameter of the circle, which is 2r.
We can use the Pythagorean theorem to find the height of each trapezoid leg. For example, for the left leg:
a² + r² = 16²/4
a² + r² = 64
Similarly, for the right leg:
b² + r² = 6²/4
b² + r² = 9
We can solve for r in each equation:
r² = 64 - a²
r² = 9 - b²
Setting the right-hand sides equal to each other:
64 - a² = 9 - b²
Simplifying:
a² + b² = 55
Now we can use the fact that the circumference of the circle (2πr) is equal to the sum of the lengths of the sides of the polygon. The sides of the polygon are the four trapezoid legs and the top and bottom sides.
The top and bottom sides have lengths 7 and 4, respectively. For the trapezoid legs, we can use the Pythagorean theorem and our equations above to find their lengths:
sqrt(a² + r²) + sqrt(b² + r²) = sqrt((16-7)²/4 + 4r²) + sqrt((6-4)²/4 + 4r²) = sqrt(27 + 4r²) + sqrt(5 + 4r²)
So the perimeter of the polygon is:
Perimeter = 7 + 4 + sqrt(27 + 4r²) + sqrt(5 + 4r²) + 2πr
We don't know the value of r yet, but we can use the equation we derived earlier:
a² + b² = 55
Substituting in r²:
(64 - r²) + (9 - r²) = 55
Solving for r:
r² = 6
Now we can substitute r² = 6 into our equation for the perimeter:
Perimeter = 7 + 4 + sqrt(27 + 24) + sqrt(5 + 24) + 2π(sqrt(6))
Perimeter = 11 + 5sqrt(3) + 2πsqrt(6)
So the perimeter of the polygon is approximately 27.9 cm.