Question

Segment AB
intersects chord MN¯¯¯¯¯¯¯¯¯¯
at a right angle at point O.

Circle C as described in the problem. Angle B O N is marked as a right angle. If MO=5x+3
and NO=6x,
what is the length of MN¯¯¯¯¯¯¯¯¯¯?
Enter the correct value.

Answer 1

To find the length of MN, we first need to find the length of ON using the given information.

Since angle BON is a right angle, we can use the Pythagorean theorem in triangle BON:
NO^2 = BO^2 + BN^2
(6x)^2 = (5x+3)^2 + (MN)^2
36x^2 = 25x^2 + 30x + 9 + MN^2
11x^2 - 30x - 9 = MN^2

To find MN, we need to find the value of x. We can use the fact that MO = 5x + 3:
5x + 3 = ON = 6x
x = 3

Now we can find the length of MN:
MN^2 = 11(3)^2 - 30(3) - 9
MN^2 = 99 - 90 - 9
MN^2 = 0
MN = 0

Therefore, the length of MN is 0.