Answer:
5 cm
Step-by-step explanation:
When two chords are equidistant from the center of the circle, the two chords have equal length. Therefore, the length of chord AB, 4x - 6, is equal to the length of chord CD, 6x - 12.
4x - 6 = 6x - 12
6 = 2x
x = 3
Now that we know x = 3, we can substitute the value back into the original expression, 4x - 6, to find the length of chord AB.
AB = 4x - 6 = 4*3 - 6 = 12 - 6 = 6 cm
When measuring the distance between a line and a point, we create a segment through the point perpendicular to the line. In Geometry, we also learn that the perpendicular bisector of a chord in a circle contains the center of the circle.
In this case, the gray line perpendicularly bisects segment AB. PX is 4 cm long and AX is 3 cm long (because AX is half the length of AB). Notice that triangle AXP is a right triangle, so we can use Pythagorean Theorem to find AP.
[tex]AP^{2} = AX^{2} +PX^{2}[/tex]
[tex]AP = \sqrt{3^{2} +4^{2} }[/tex]
[tex]AP =\sqrt{9+16}[/tex]
[tex]AP = \sqrt{25}[/tex]
[tex]AP = 5[/tex]
