Question

give the new coordinates for dilating graph the image of the rectangle below using a scale factor of k = \\(\frac{5}{2}\\) image of grid with rectangle write the numerical answer (ie if you get 2 for your answer, type \2\ not \two\)

Explanation:

To solve the problem of dilating the rectangle with a scale factor \( k = \frac{5}{2} \), we first need the original coordinates of the rectangle's vertices. Let's assume the original coordinates of the vertices (from the graph) are:

• \( J(x_1, y_1) \)
• \( K(x_2, y_2) \)
• \( L(x_3, y_3) \)
• \( M(x_4, y_4) \)

Step 1: Identify Original Coordinates

From the grid, let's determine the original coordinates (assuming each grid square is 1 unit):

• Let's assume \( J(-4, 2) \), \( K(-2, 4) \), \( L(2, 0) \), \( M(0, -2) \) (we need to confirm the exact coordinates from the graph, but let's proceed with typical grid coordinates for a rectangle).

Step 2: Apply Dilation Formula

The dilation formula for a point \( (x, y) \) with scale factor \( k \) is \( (kx, ky) \).

For Point \( J(-4, 2) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times (-4) = -10 \\ y' &= \frac{5}{2} \times 2 = 5 \\ \text{New } J' &= (-10, 5) \end{align*}$$

\]

For Point \( K(-2, 4) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times (-2) = -5 \\ y' &= \frac{5}{2} \times 4 = 10 \\ \text{New } K' &= (-5, 10) \end{align*}$$

\]

For Point \( L(2, 0) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times 2 = 5 \\ y' &= \frac{5}{2} \times 0 = 0 \\ \text{New } L' &= (5, 0) \end{align*}$$

\]

For Point \( M(0, -2) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times 0 = 0 \\ y' &= \frac{5}{2} \times (-2) = -5 \\ \text{New } M' &= (0, -5) \end{align*}$$

\]

Step 3: Graph the New Coordinates

Plot the points \( J'(-10, 5) \), \( K'(-5, 10) \), \( L'(5, 0) \), and \( M'(0, -5) \) on the coordinate plane and connect them to form the dilated rectangle.

Final Answer

The new coordinates after dilation with \( k = \frac{5}{2} \) are:

• \( J'(-10, 5) \)
• \( K'(-5, 10) \)
• \( L'(5, 0) \)
• \( M'(0, -5) \)

(Note: The exact coordinates depend on the original coordinates of the rectangle from the graph. If the original coordinates are different, adjust the calculations accordingly.)

Answer:

To solve the problem of dilating the rectangle with a scale factor \( k = \frac{5}{2} \), we first need the original coordinates of the rectangle's vertices. Let's assume the original coordinates of the vertices (from the graph) are:

• \( J(x_1, y_1) \)
• \( K(x_2, y_2) \)
• \( L(x_3, y_3) \)
• \( M(x_4, y_4) \)

Step 1: Identify Original Coordinates

From the grid, let's determine the original coordinates (assuming each grid square is 1 unit):

• Let's assume \( J(-4, 2) \), \( K(-2, 4) \), \( L(2, 0) \), \( M(0, -2) \) (we need to confirm the exact coordinates from the graph, but let's proceed with typical grid coordinates for a rectangle).

Step 2: Apply Dilation Formula

The dilation formula for a point \( (x, y) \) with scale factor \( k \) is \( (kx, ky) \).

For Point \( J(-4, 2) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times (-4) = -10 \\ y' &= \frac{5}{2} \times 2 = 5 \\ \text{New } J' &= (-10, 5) \end{align*}$$

\]

For Point \( K(-2, 4) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times (-2) = -5 \\ y' &= \frac{5}{2} \times 4 = 10 \\ \text{New } K' &= (-5, 10) \end{align*}$$

\]

For Point \( L(2, 0) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times 2 = 5 \\ y' &= \frac{5}{2} \times 0 = 0 \\ \text{New } L' &= (5, 0) \end{align*}$$

\]

For Point \( M(0, -2) \):

\[

$$\begin{align*} x' &= \frac{5}{2} \times 0 = 0 \\ y' &= \frac{5}{2} \times (-2) = -5 \\ \text{New } M' &= (0, -5) \end{align*}$$

\]

Step 3: Graph the New Coordinates

Plot the points \( J'(-10, 5) \), \( K'(-5, 10) \), \( L'(5, 0) \), and \( M'(0, -5) \) on the coordinate plane and connect them to form the dilated rectangle.

Final Answer

The new coordinates after dilation with \( k = \frac{5}{2} \) are:

• \( J'(-10, 5) \)
• \( K'(-5, 10) \)
• \( L'(5, 0) \)
• \( M'(0, -5) \)

(Note: The exact coordinates depend on the original coordinates of the rectangle from the graph. If the original coordinates are different, adjust the calculations accordingly.)