Question

finding the length of a chord
circle j is congruent to circle p.
if $ok = 2x$, $qr = 12$, and $rt = x + 3$, what is the length of chord km?
8
9
10
12

Explanation:

Step1: Set equal congruent chords

Since circles J and P are congruent, chords subtending congruent central angles are equal. So $OK = RT$.
$2x = x + 3$

Step2: Solve for x

Subtract $x$ from both sides.
$2x - x = 3 \implies x = 3$

Step3: Find OK length

Substitute $x=3$ into $OK=2x$.
$OK = 2\times3 = 6$

Step4: Find KM length

Chord $KM$ is congruent to chord $QR$ (they subtend congruent central angles in congruent circles), and $QR=12$. Wait, correction: Wait, no—wait, $OK$ is a radius, and $KM$ is a chord. Wait, no, actually, since $\angle OKM$'s central angle matches $\angle QRT$? No, correct: Congruent circles have equal radii, so $OK=QJ=PR=PT$, and chords $KM$ and $QR$ are congruent? No, wait, no: The central angles for $KM$ and $QR$ are marked as congruent, so chords $KM=QR$? No, wait no, the given: $OK=2x$, $RT=x+3$, and $OK=RT$ (radii of congruent circles are equal). Then $x=3$, so $OK=6$. But $QR=12$, and since central angles for $KM$ and $QR$ are congruent, chord length $KM=QR=12$? No, wait no, wait: Wait, $RT$ is a radius, so $RT=OK$ (radii of congruent circles). So $2x=x+3$, $x=3$, so radius is 6. Then $QR=12$, which is twice the radius? That can't be, a chord can't be longer than diameter. Wait, diameter is 12, so $QR$ is the diameter, so $KM$ is also a diameter, so $KM=12$. Wait, no, wait, $QR=12$, if radius is 6, diameter is 12, so $QR$ is diameter, so $KM$ is also diameter, so $KM=12$. Wait, but let's recheck:

Wait, correct logic: Congruent circles have equal radii, so $OK$ (radius of J) = $RT$ (radius of P). So $2x=x+3$, solve for x: $x=3$. Then radius is 6. Now, chord $QR=12$, which is equal to the diameter ($2\times6=12$), so $QR$ is a diameter. The central angle for $KM$ is congruent to the central angle for $QR$, so $KM$ must also be a diameter, so $KM=12$.

Wait, no, wait another way: If two chords in congruent circles subtend congruent central angles, they are equal. So $KM=QR=12$. But we confirm the radius is 6, so diameter is 12, which matches.

Answer:

12