Question

find the radius of \\(\odot q\\).
the radius of circle \\(q\\) is \\(\square\\) units.

Explanation:

Step1: Set up chord distance equality

Since chords $EF$ and $HG$ are congruent (both length 16), their distances from the center $Q$ are equal:
$4x + 3 = 7x - 6$

Step2: Solve for $x$

Rearrange to isolate $x$:
$3 + 6 = 7x - 4x$
$9 = 3x$
$x = \frac{9}{3} = 3$

Step3: Calculate radius with Pythagoras

Let $r$ be the radius. Use half the chord length ($\frac{16}{2}=8$) and distance from center ($4x+3$):
$r^2 = 8^2 + (4x+3)^2$
Substitute $x=3$:
$r^2 = 64 + (4(3)+3)^2 = 64 + 15^2 = 64 + 225 = 289$
$r = \sqrt{289} = 17$

Answer:

17