15 POINTS!!!!!!!
1. Quadrilateral OPQR is inscribed inside a circle as shown below. What is the measure of angle O? You must show all work and calculations to receive credit. (10 points) Circle N is shown with an inscribed quadrilateral labeled OPQR. O is labeled 2x degrees. P is labeled y degrees. Q is labeled 2

2. Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.

x2 + 2x + y2 + 4y = 20 (10 points)

Answers

1. For any inscribed quadrilateral, the sum of each pair of opposite angles must add up to 180 degrees. Therefore, O + Q = 180 2x + (2x + 4) = 180 4x + 4 = 180 4x = 176 x = 44 degrees angle O = 2x = 88 degrees. 2. To find the center and radius: x2 + 2x + y2 + 4y = 20 (x^2 + 2x + 1) - 1 + (y^2 + 4y + 4) - 4 = 20 (x + 1)^2 + (y + 2)^2 - 5 = 20 (x + 1)^2 + (y + 2)^2 = 25 (x + 1)^2 + (y + 2)^2 = 5^2 Therefore, the center is at x = -1, y = -2 , or (-1, -2). The radius is 5 units.


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