Question

1. what are the coordinates of a point that maps back to itself under a dilation centered at the origin? explain.

1. consider △pqr in the coordinate plane.

graph of △pqr on a coordinate grid with p, q, r labeled

a. what are the coordinates of the images of vertices p, q, and r under a dilation centered at the origin with scale factor 1.5?

b. graph and label the image of △pqr under the same dilation.

Explanation:

Question 5

Step1: Recall Dilation Formula

A dilation centered at the origin with scale factor \( k \) transforms a point \( (x, y) \) to \( (kx, ky) \). For a point to map to itself, \( (x, y) = (kx, ky) \).

Step2: Solve for \( x, y \)

This implies \( x = kx \) and \( y = ky \). Rearranging, \( x(1 - k) = 0 \) and \( y(1 - k) = 0 \). If \( k
eq 1 \), then \( x = 0 \) and \( y = 0 \). If \( k = 1 \), any point maps to itself, but the question is about a general dilation (implied by "maps back to itself under a dilation" – the only point that works for any dilation is the origin). So the point is \( (0, 0) \).

Step1: Find Original Coordinates

From the graph:

• Point \( P \): Let's assume the grid has 1 - unit squares. Looking at the graph, \( P \) is at \( (-5, 2) \).
• Point \( Q \): \( Q \) is at \( (-3, 4) \).
• Point \( R \): \( R \) is at \( (-1, 4) \).

Step2: Apply Dilation Formula

Dilation centered at the origin with scale factor \( 1.5 \) (or \( \frac{3}{2} \)): For a point \( (x, y) \), the image is \( (1.5x, 1.5y) \).

• For \( P(-5, 2) \): \( x'=1.5\times(-5)=-7.5 \), \( y' = 1.5\times2 = 3 \). So \( P'(-7.5, 3) \).
• For \( Q(-3, 4) \): \( x'=1.5\times(-3)=-4.5 \), \( y' = 1.5\times4 = 6 \). So \( Q'(-4.5, 6) \).
• For \( R(-1, 4) \): \( x'=1.5\times(-1)=-1.5 \), \( y' = 1.5\times4 = 6 \). So \( R'(-1.5, 6) \).

Step1: Plot Image Points

• For \( P'(-7.5, 3) \): Move \( 7.5 \) units to the left of the origin on the \( x \) - axis and \( 3 \) units up on the \( y \) - axis.
• For \( Q'(-4.5, 6) \): Move \( 4.5 \) units to the left of the origin on the \( x \) - axis and \( 6 \) units up on the \( y \) - axis.
• For \( R'(-1.5, 6) \): Move \( 1.5 \) units to the left of the origin on the \( x \) - axis and \( 6 \) units up on the \( y \) - axis.

Step2: Draw the Triangle

Connect the points \( P' \), \( Q' \), and \( R' \) to form the image of \( \triangle PQR \) under the dilation. Label the points \( P' \), \( Q' \), \( R' \) respectively.

Answer:

The coordinates are \( (0, 0) \) (the origin), because a dilation centered at the origin with scale factor \( k \) maps \( (x, y) \) to \( (kx, ky) \); for \( (x, y) = (kx, ky) \) to hold for the dilation (regardless of \( k \), or for a specific non - identity dilation), \( x = 0 \) and \( y = 0 \).

Question 6a