Question

dilate figure pqrs by a scale factor of \\(\frac{1}{2}\\) with the center of dilation at point p. \\(qp = 4\\) units \\(sp = 3\\) units \\(rs = 2\\) units move the points along the rays to plot points \\(r\\), \\(s\\), and \\(t\\).

Explanation:

Step1: Recall Dilation Rules

Dilation with center \( P \) and scale factor \( \frac{1}{2} \) means each point \( X \) (other than \( P \)) will be mapped to \( X' \) such that \( PX'=\frac{1}{2}PX \) and \( X' \) lies on the ray \( PX \).

Step2: Analyze Point \( Q \)

Let's assume original \( PQ \) length. After dilation, \( PQ'=\frac{1}{2}PQ \). Given \( Q'P = 4 \) units (wait, maybe original \( PQ \): if \( Q'P=\frac{1}{2}PQ \), then \( PQ = 2\times Q'P=8 \) units? Wait, no, the problem says "Dilate figure \( PQRS \) by a scale factor of \( \frac{1}{2} \) with center at \( P \)". So for each point:

- For point \( Q \): The ray is \( PQ \). So \( Q' \) is on \( PQ \) such that \( PQ'=\frac{1}{2}PQ \).
- For point \( S \): The ray is \( PS \). \( S' \) is on \( PS \) such that \( PS'=\frac{1}{2}PS \). Given \( S'P = 3 \) units (so \( PS' = 3 \), so original \( PS = 6 \) units? Wait, maybe the given \( Q'P = 4 \), \( S'P = 3 \), \( R'S' = 2 \) are after dilation. Wait, the task is to plot \( R', S', T \) (wait, maybe a typo, maybe \( Q', R', S' \)) by moving along rays from \( P \).

Step3: Plotting \( S' \)

Ray from \( P \) through \( S \): since scale factor \( \frac{1}{2} \), \( S' \) is halfway between \( P \) and \( S \) (because scale factor \( \frac{1}{2} \) means distance from \( P \) is half of original). If original \( PS \) is, say, from \( P \) (origin? Let's assume grid: \( P \) is at bottom - left, \( S \) is to the right. So moving from \( P \) towards \( S \), \( S' \) is at \( \frac{1}{2} \) the distance of \( PS \).

Step4: Plotting \( R' \)

Ray from \( P \) through \( R \): \( R' \) is on ray \( PR \) such that \( PR'=\frac{1}{2}PR \). Also, \( R'S' \) should be \( \frac{1}{2}RS \) (since dilation preserves shape, so sides are scaled by \( \frac{1}{2} \)). Given \( R'S' = 2 \) units, so original \( RS = 4 \) units.

Step5: Plotting \( Q' \)

Ray from \( P \) through \( Q \): \( Q' \) is on ray \( PQ \) with \( PQ'=\frac{1}{2}PQ \). Given \( Q'P = 4 \) units, so that's the distance from \( P \) to \( Q' \), so \( Q' \) is 4 units from \( P \) along \( PQ \).

To plot:
- For \( S' \): Start at \( P \), move along the ray (horizontal, since \( PS \) is horizontal) half the distance to \( S \). If original \( PS \) is, say, 6 units (since \( S'P = 3 \)), then \( S' \) is 3 units from \( P \) towards \( S \).
- For \( R' \): Start at \( P \), move along ray \( PR \) (the dashed line) half the distance to \( R \). Then, from \( R' \), move vertically (since \( RS \) is vertical) 2 units (since \( R'S' = 2 \)) to align with \( S' \) horizontally? Wait, maybe better to use the scale factor: each coordinate (if \( P \) is at \( (0,0) \), \( Q \) at \( (0,8) \), \( S \) at \( (6,0) \), \( R \) at \( (6,4) \)):

- Dilation of \( Q(0,8) \) with center \( P(0,0) \) and scale \( \frac{1}{2} \): \( Q'(0, 8\times\frac{1}{2})=(0,4) \) (so \( Q'P = 4 \) units, matches).
- Dilation of \( S(6,0) \) with center \( P(0,0) \) and scale \( \frac{1}{2} \): \( S'(3,0) \) (so \( S'P = 3 \) units, matches).
- Dilation of \( R(6,4) \) with center \( P(0,0) \) and scale \( \frac{1}{2} \): \( R'(3,2) \). Then \( R'S' \): distance between \( (3,2) \) and \( (3,0) \) is 2 units, which matches \( R'S' = 2 \) units.

So to plot:
- \( Q' \): 4 units up from \( P \) along \( PQ \) (vertical ray).
- \( S' \): 3 units right from \( P \) along \( PS \) (horizontal ray).
- \( R' \): 3 units right and 2 units up from \( P \) (along ray \( PR \)), or from \( S' \), 2 units up, and from \( Q' \), 3 units right and 2 units down? Wait, using the coordinates, \( R' \) is at \( (3,2) \), so from \( P(0,0) \), move 3 right (x - direction) and 2 up (y - direction) along ray \( PR \) (which goes from \( P(0,0) \) to \( R(6,4) \), so slope \( \frac{4}{6}=\frac{2}{3} \), so moving 3 right (x=3) gives y=2, which is correct).

Answer:

To plot the dilated points:

• \( Q' \): Move 4 units along the vertical ray from \( P \) (toward \( Q \)).
• \( S' \): Move 3 units along the horizontal ray from \( P \) (toward \( S \)).
• \( R' \): Move along the ray from \( P \) through \( R \) until it is halfway (scale factor \( \frac{1}{2} \)) between \( P \) and \( R \). From \( S' \), move 2 units up (since \( R'S' = 2 \) units) to reach \( R' \), or use coordinates: if \( P=(0,0) \), \( R'=(3,2) \) (3 units right, 2 units up from \( P \)).

(Note: The exact plotting depends on the grid, but the key is using the scale factor \( \frac{1}{2} \) along the rays from \( P \) for each vertex.)