+23 1. Let O be the center of a circle with diameter AB. Let C be a point on the circle such that angle COA = 90. Point P lies on line segment OA, and when line segment CP is extended past P, it intersects the circle at Q. If PQ = 7 and PO = 20, then find r2 where r is the radius of the circle.
2. The diagram shows two chords that are parallel. The lengths of the chords are 18 and 14 and the distance between the two chords is 8. What is the length of the chord that lies halfway between them?
1. First, we notice that triangle OCP is a right triangle with hypotenuse OP and angle OCP = 90 degrees. Therefore, we can use the Pythagorean theorem to find CP:
CP^2 = OP^2 - OC^2 = 20^2 - r^2
Next, we notice that triangle CPQ is similar to triangle COA (by angle-angle similarity), so we have:
CP/QP = CO/OA
Substituting CP from the previous equation and simplifying, we get:
(20^2 - r^2)/QP = r/(2r)
Simplifying this equation and solving for QP, we get:
QP = 40r/(20^2 - r^2)
We also know that PQ = 7, so we can set up another equation:
PQ^2 + QO^2 = PO^2
Substituting PQ and QP from the previous equations and simplifying, we get:
7^2 + (40r/(20^2 - r^2) - r)^2 = 20^2
Expanding and simplifying this equation, we get a quadratic equation in r^2:
r^4 - 400r^2 + 9600 = 0
Using the quadratic formula, we get:
r^2 = (400 ± √(400^2 - 4*9600))/2 = 200 ± 20√5
Since r is the radius of the circle, we know that r > 0, so we take the positive square root:
r^2 = 200 + 20√5
Therefore, the answer is 200 + 20√5.
