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 ( overline{km} )?
options: 9, 12, 8

Explanation:

Step1: Recall Congruent Circles Property

Congruent circles have equal radii, so \( OK = QR \) (radii of congruent circles \( J \) and \( P \)). Given \( OK = 2x \) and \( QR = 12 \), set \( 2x = 12 \).

Step2: Solve for \( x \)

From \( 2x = 12 \), divide both sides by 2: \( x = \frac{12}{2} = 6 \).

Step3: Find \( RT \)

Given \( RT = x + 3 \), substitute \( x = 6 \): \( RT = 6 + 3 = 9 \). Wait, no—wait, in congruent circles, chords equidistant from center or with equal central angles? Wait, actually, since circles are congruent, and \( KM \) and \( QT \) (or \( RT \) related? Wait, no, the diagram: in circle \( P \), \( QR \) is a radius? Wait, no, \( Q \), \( T \), \( R \): \( P \) is center, so \( PQ \), \( PR \), \( PT \) are radii. So \( QR \) is a chord? Wait, no, the problem: \( OK = 2x \) (radius of circle \( J \)), \( QR = 12 \) (chord? No, \( Q \) and \( R \) on circle \( P \), \( P \) is center, so \( PQ = PR = PT \) (radii). Wait, maybe \( QR \) is a chord, but \( RT \) is a segment. Wait, correction: Since circles \( J \) and \( P \) are congruent, their radii are equal. So \( OK \) (radius of \( J \)) equals \( PQ \) (radius of \( P \)). Then, in circle \( P \), \( RT \) is a segment, but maybe \( KM \) and \( QT \) are chords? Wait, no, the question is length of chord \( KM \). Wait, maybe \( KM \) is equal to \( QT \), and \( QT = 2 \times RT \)? Wait, no, let's re-express. Wait, the key: \( OK = 2x \), \( QR = 12 \), and since circles are congruent, \( OK = QR \) (radii), so \( 2x = 12 \) gives \( x = 6 \). Then \( RT = x + 3 = 9 \). But wait, maybe \( KM = 2 \times RT \)? No, that doesn't make sense. Wait, no—wait, maybe \( KM \) is equal to \( QR \)? No, \( QR = 12 \), but the options have 12. Wait, maybe I messed up. Wait, the problem says "what is the length of chord \( \overline{KM} \)". Since circles are congruent, and if \( KM \) and \( QR \) are chords with equal central angles, or since \( OK \) is radius, \( KM \) is a chord. Wait, maybe the correct approach: \( OK = 2x \), \( QR = 12 \), so \( 2x = 12 \) (since radii of congruent circles are equal), so \( x = 6 \). Then \( RT = x + 3 = 9 \), but that's not. Wait, no, maybe \( KM = QR \)? Because circles are congruent, so chords with equal radii and same central angle? Wait, the diagram: in circle \( J \), \( K \) and \( M \) on circle, \( J \) is center, so \( JK = JM = OK \) (radii). In circle \( P \), \( Q \), \( T \), \( R \) on circle, \( P \) is center, so \( PQ = PR = PT \). The problem: \( OK = 2x \), \( QR = 12 \), \( RT = x + 3 \). Wait, maybe \( KM \) is equal to \( QT \), and \( QT = 2 \times RT \)? No, \( RT = 9 \), \( 2 \times 9 = 18 \), not an option. Wait, the options are 9, 12, 8. Wait, maybe \( KM = QR \)? \( QR = 12 \), so answer is 12. Because circles are congruent, so chords with equal radii and same central angle (or since \( OK \) is radius, \( KM \) is a chord, and \( QR \) is a chord of equal length? Wait, maybe the key is that \( KM = QR \) because the circles are congruent and the chords are corresponding. So since \( QR = 12 \), \( KM = 12 \).

Answer:

12