Chords GH and JK in circle O intersect at W. If GW= x, WH= x+8, JW= 6, and KW= 8, find the value of x and the length of GH
First, we can use the Power of a Point theorem to relate the lengths of GW, WH, JW, and KW:
GW * WH = JW * KW
x * (x+8) = 6 * 8
x^2 + 8x = 48
x^2 + 8x - 48 = 0
We can solve for x using the quadratic formula:
x = (-8 ± sqrt(8^2 - 4*1*(-48))) / (2*1)
x = (-8 ± sqrt(256)) / 2
x = (-8 ± 16) / 2
x = 4 or x = -12
Since x represents a length, we can disregard the negative solution and conclude that x = 4.
To find the length of GH, we can use the Pythagorean theorem on right triangles GWH and JWK:
GH^2 = (GW + WH)^2 = (4 + 12)^2 = 256
GH = sqrt(256) = 16
Therefore, the value of x is 4 and the length of GH is 16.