Question

problem 5. here is rectangle pqrs. draw the image of rectangle pqrs after the dilation. a dilation using point r as the center of dilation and a scale factor of 2.

Explanation:

Step1: Recall dilation formula

For a dilation with center \(C(x_c,y_c)\) and scale - factor \(k\), the image of a point \(P(x,y)\) is given by \(P'(x_c + k(x - x_c),y_c + k(y - y_c))\). Here, the center of dilation \(C = R\), let's assume \(R\) has coordinates \((x_R,y_R)\), and \(k = 2\).

Step2: Dilate point \(P\)

Let the coordinates of \(P\) be \((x_P,y_P)\). Then the coordinates of \(P'\) are \((x_R+2(x_P - x_R),y_R + 2(y_P - y_R))=(2x_P - x_R,2y_P - y_R)\).

Step3: Dilate point \(Q\)

Let the coordinates of \(Q\) be \((x_Q,y_Q)\). Then the coordinates of \(Q'\) are \((x_R+2(x_Q - x_R),y_R + 2(y_Q - y_R))=(2x_Q - x_R,2y_Q - y_R)\).

Step4: Dilate point \(S\)

Let the coordinates of \(S\) be \((x_S,y_S)\). Then the coordinates of \(S'\) are \((x_R+2(x_S - x_R),y_R + 2(y_S - y_R))=(2x_S - x_R,2y_S - y_R)\).

Step5: Plot the new rectangle

Connect the dilated points \(P'\), \(Q'\), \(R\) (since the center of dilation \(R\) remains the same), and \(S'\) to form the dilated rectangle.

Since this is a geometric construction problem on a grid, we can also use a non - coordinate approach:

Step1: Measure distances from \(R\)

Measure the horizontal and vertical distances from the center of dilation \(R\) to points \(P\), \(Q\), and \(S\).

Step2: Double the distances

Multiply these distances by the scale factor \(2\).

Step3: Locate new points

Starting from \(R\), move the new (doubled) distances in the appropriate directions to locate the dilated points \(P'\), \(Q'\), and \(S'\).

Step4: Draw the dilated rectangle

Connect the points \(P'\), \(Q'\), \(R\), and \(S'\) to form the dilated rectangle.

In a grid - based system, count the number of grid - squares from \(R\) to the other points, double those counts, and mark the new points on the grid to draw the dilated rectangle.

Answer:

To draw the image, first, measure the distances from point \(R\) to points \(P\), \(Q\), and \(S\) in terms of horizontal and vertical grid - units. Then, starting from \(R\), move twice the measured distances in the same directions to find the new positions of \(P\), \(Q\), and \(S\) (denoted as \(P'\), \(Q'\), \(S'\)). Since \(R\) is the center of dilation, it remains in the same position. Connect \(P'\), \(Q'\), \(R\), and \(S'\) to form the dilated rectangle.