Question

1
select the correct location on the table.
segment lm is a dilation centered at point p of segment jk.

which expression represents the scale factor of the dilation?

$lm - jk$	$\frac{1}{2}(pj + pk)$	$\frac{mp}{kp}$
$mp - pl$	$\frac{1}{2}(jk + lm)$	$\frac{lj}{jp}$

Explanation:

Step1: Recall Dilation Scale Factor

The scale factor of a dilation centered at a point is the ratio of the length of a segment from the center to a point on the image to the length of the corresponding segment from the center to the pre - image point. For dilation centered at \(P\), if we consider the segments along the lines from \(P\), for the lines \(PM\) and \(PK\) (or \(PL\) and \(PJ\)), the scale factor \(k=\frac{\text{length of image segment from }P}{\text{length of pre - image segment from }P}\).

Step2: Analyze the Options

• Option \(\frac{MP}{KP}\): Since \(LM\) is the image of \(JK\) under dilation centered at \(P\), the points \(M\) and \(K\) are corresponding points, and \(L\) and \(J\) are corresponding points. The distance from \(P\) to \(M\) (i.e., \(MP\)) and the distance from \(P\) to \(K\) (i.e., \(KP\)) are along the same line (the line \(PM\) (or \(PK\))). So the ratio \(\frac{MP}{KP}\) gives the scale factor because it is the ratio of the length of the segment from the center of dilation \(P\) to the image point \(M\) to the length of the segment from the center of dilation \(P\) to the pre - image point \(K\).
• Let's check other options:
• \(LM - JK\): This is a difference of lengths of the image and pre - image segments, not a ratio, so it can't be the scale factor.
• \(\frac{1}{2}(PJ + PK)\): This is an average - like expression of two lengths, not a ratio of image to pre - image from the center, so it's not the scale factor.
• \(MP - PL\): This is a difference of lengths, not a ratio, so it's not the scale factor.
• \(\frac{1}{2}(JK + LM)\): This is an average of the lengths of the pre - image and image segments, not a ratio from the center, so it's not the scale factor.
• \(\frac{LJ}{JP}\): \(LJ\) and \(JP\) are not corresponding segments in the dilation in the correct ratio sense. \(LJ\) is \(PL - PJ\) and \(JP\) is \(PJ\), and this ratio does not represent the scale factor of the dilation of \(JK\) to \(LM\).

Answer:

\(\frac{MP}{KP}\) (the third option in the first row of the table)