Question

13 numeric 1 point
if ( pq = qr ), ( overline{jk}=3x + 23 ) and ( lm = 9x - 19 ), find ( pk ).
answer

Explanation:

Step1: Use the property of chords

Since \(PQ = QR\), the chords \(JK\) and \(LM\) are equidistant from the center \(Q\) of the circle. By the property of circles, chords that are equidistant from the center are equal in length. So, \(JK=LM\).
$$3x + 23=9x-19$$

Step2: Solve for \(x\)

Subtract \(3x\) from both sides:
$$23 = 9x-3x - 19$$
$$23=6x - 19$$
Add \(19\) to both sides:
$$23 + 19=6x$$
$$42=6x$$
Divide both sides by \(6\):
$$x = 7$$

Step3: Find the length of \(JK\)

Substitute \(x = 7\) into the expression for \(JK\): \(JK=3x + 23\)
$$JK=3\times7+23=21 + 23=44$$
Since \(OQ\perp JK\), by the perpendicular - from - the - center - to - a - chord theorem, \(PR\) bisects \(JK\). So \(PK=\frac{1}{2}JK\)
$$PK=\frac{44}{2}=22$$

Answer:

\(22\)