Segment JK

is a diameter of circle G

and intersects EF¯¯¯¯¯¯¯¯

at a right angle at point P.
JK¯¯¯¯¯¯¯¯

intersects WV¯¯¯¯¯¯¯¯¯

at a right angle at point O.
Circle G as described in the problem. Angle F P G and angle V O K are marked as right angles. If PG=x−4

and GO=12x+3,

what is the length of PG¯¯¯¯¯¯¯¯?

Responses
14

14
10

10
7

7
−1

1 answer

To find the length of PG, we can set up an equation using the Pythagorean theorem since a diameter of a circle is twice the radius.

Consider triangle PGF. We have:
(FP)^2 + (PG)^2 = (FG)^2

Since PG is the diameter, FG = 2(PG). So we have:
(FP)^2 + (PG)^2 = (2PG)^2
(FP)^2 + (PG)^2 = 4(PG)^2
Given that FP is x-4 and PG is x, we can substitute these values in:
(x-4)^2 + x^2 = 4x^2
x^2 - 8x + 16 + x^2 = 4x^2
2x^2 - 8x + 16 = 4x^2
2x^2 - 8x + 16 - 4x^2 = 0
-2x^2 - 8x + 16 = 0

Now we need to solve this quadratic equation to find the value of x. The length of PG is then x. Let's use the quadratic formula:
x = (-(-8) ± √((-8)^2 - 4*(-2)*16))/(2*(-2))
x = (8 ± √(64 + 128))/(-4)
x = (8 ± √192)/-4
x = (8 ± 4√3)/-4
x = -2 ± √3

Now x could be either of the roots, but in the context of the problem, length cannot be negative. So the length of PG would be x = -2 + √3. Approximating this gives us around 0.73.

Therefore, the length of PG is approximately 0.73. The closest whole number to this would be 1.

Hence, the correct response is 1.