the circle is circumscribed by the pentagon as shown (not drawn to scale). If QZ=10, YX=9, XW=9, UW=17, and SU=10, find the perimeter of the pentagon.

epic style

To find the perimeter of the pentagon, we need to determine the lengths of the remaining sides. Let's go step by step.

1. Start by drawing the pentagon and the circle inscribed within it.

2. Label the points of intersection between the circle and the pentagon as A, B, C, D, and E, clockwise from the top.

3. Since the circle is circumscribed around the pentagon, the radii of the circle will be the perpendicular distance from the center of the circle to the sides of the pentagon. Denote the center of the circle as O.

4. Draw the radii OZ, OY, OX, OW, and OU.

5. According to the problem, QZ = 10, YX = 9, XW = 9, UW = 17, and SU = 10.

Now, let's use these given lengths to find the remaining side lengths.

6. The pentagon consists of five sides, so we need to find two more sides.

7. Notice that triangle QZO is an isosceles triangle because QZ is tangent to the circle at point Z, and OZ is a radius. Therefore, triangle QZO is also a right triangle since it is isosceles.

8. We can use the Pythagorean theorem to find OZ, which is the radius of the circle.

QZ^2 = OZ^2 + OQ^2
10^2 = OZ^2 + OQ^2
OZ^2 = 100 - OQ^2

9. Triangle YXO is an isosceles triangle because YX is tangent to the circle at point Y, and OX is a radius. Therefore, triangle YXO is also a right triangle since it is isosceles.

10. We can use the Pythagorean theorem to find OY.

YX^2 = OY^2 + OX^2
9^2 = OY^2 + OX^2
OY^2 = 81 - OX^2

11. Using the same reasoning, we can find the length of OX.

XW^2 = OX^2 + OW^2
9^2 = OX^2 + OW^2
OW^2 = 81 - OX^2

12. Triangle UWO is a right triangle, and we can again use the Pythagorean theorem to find OU.

UW^2 = OU^2 + OW^2
17^2 = OU^2 + OW^2
OU^2 = 289 - OW^2

13. Triangle SUO is a right triangle, and we can use the Pythagorean theorem to find SO.

SU^2 = SO^2 + OU^2
10^2 = SO^2 + OU^2
SO^2 = 100 - OU^2

14. Now we have the values of OZ, OY, OX, OW, and SO.

15. The perimeter of the pentagon can be calculated by adding up all the side lengths.

Pentagon Perimeter = QZ + ZO + OY + YX + XW + WO + OU + US

16. Substitute the given values and the calculated values into the perimeter formula.

Pentagon Perimeter = 10 + OZ + OY + 9 + 9 + OW + OU + 10

17. Substitute the expression for OZ, OY, OW, and OU derived in Steps 8, 9, 10, and 12.

Pentagon Perimeter = 10 + sqrt(100 - OQ^2) + sqrt(81 - OX^2) + 9 + 9 + sqrt(81 - OX^2) + sqrt(289 - OW^2) + 10

18. Substitute the expression for OX derived in Step 10.

Pentagon Perimeter = 10 + sqrt(100 - OQ^2) + sqrt(81 - OX^2) + 9 + 9 + sqrt(81 - (81 - OX^2)) + sqrt(289 - (81 - OX^2)) + 10

19. Simplify the equation and solve for the remaining variables, if possible.

Note: The exact values of the variables OQ and OX are not provided in the question. We need additional information to solve for the perimeter of the pentagon.

Please type in the subject, not grade.

We have no way of knowing what is shown.