Question

fill in the blank 6 points
give the new coordinates for dilating graph the image of the rectangle below using
a scale factor of k = \frac{5}{2}
grid image of 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 (let's call them \( J, A, L, M \)) are, for example, \( J(x_1, y_1) \), \( A(x_2, y_2) \), \( L(x_3, y_3) \), \( M(x_4, y_4) \). However, since the image is not fully clear, we'll use a general approach.

Step 1: Recall the Dilation Formula

When dilating a point \( (x, y) \) with a scale factor \( k \) centered at the origin (assuming the dilation is centered at the origin, which is common if not specified otherwise), the new coordinates \( (x', y') \) are given by:
\[
(x', y') = (k \cdot x, k \cdot y)
\]

Step 2: Identify Original Coordinates (Example)

Suppose the original coordinates of the rectangle’s vertices are:

• \( J(2, 2) \)
• \( A(4, 4) \)
• \( L(8, 2) \)
• \( M(6, 0) \)

(Note: These are example coordinates; adjust based on the actual graph. For a grid, count the units from the axes.)

Step 3: Apply the Dilation Factor \( k = \frac{5}{2} \)

For each vertex, multiply the \( x \)- and \( y \)-coordinates by \( \frac{5}{2} \):

• For \( J(2, 2) \):

\( x' = \frac{5}{2} \cdot 2 = 5 \), \( y' = \frac{5}{2} \cdot 2 = 5 \) → New \( J'(5, 5) \)

• For \( A(4, 4) \):

\( x' = \frac{5}{2} \cdot 4 = 10 \), \( y' = \frac{5}{2} \cdot 4 = 10 \) → New \( A'(10, 10) \)

• For \( L(8, 2) \):

\( x' = \frac{5}{2} \cdot 8 = 20 \), \( y' = \frac{5}{2} \cdot 2 = 5 \) → New \( L'(20, 5) \)

• For \( M(6, 0) \):

\( x' = \frac{5}{2} \cdot 6 = 15 \), \( y' = \frac{5}{2} \cdot 0 = 0 \) → New \( M'(15, 0) \)

Step 4: Verify with Actual Graph

If the original coordinates differ (e.g., from the grid), adjust the calculation. For example, if a vertex is at \( (x, y) \), multiply both by \( \frac{5}{2} \) to get the dilated coordinates.

Final Answer (Example)

If the original vertex was \( (2, 2) \), the dilated coordinate is \( (5, 5) \). Adjust based on the actual original coordinates from the graph.

(Note: Replace with the correct coordinates from the problem’s graph. For instance, if the original \( J \) is at \( (2, 2) \), the answer for \( J' \) is \( (5, 5) \), etc.)

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 (let's call them \( J, A, L, M \)) are, for example, \( J(x_1, y_1) \), \( A(x_2, y_2) \), \( L(x_3, y_3) \), \( M(x_4, y_4) \). However, since the image is not fully clear, we'll use a general approach.

Step 1: Recall the Dilation Formula

When dilating a point \( (x, y) \) with a scale factor \( k \) centered at the origin (assuming the dilation is centered at the origin, which is common if not specified otherwise), the new coordinates \( (x', y') \) are given by:
\[
(x', y') = (k \cdot x, k \cdot y)
\]

Step 2: Identify Original Coordinates (Example)

Suppose the original coordinates of the rectangle’s vertices are:

• \( J(2, 2) \)
• \( A(4, 4) \)
• \( L(8, 2) \)
• \( M(6, 0) \)

(Note: These are example coordinates; adjust based on the actual graph. For a grid, count the units from the axes.)

Step 3: Apply the Dilation Factor \( k = \frac{5}{2} \)

For each vertex, multiply the \( x \)- and \( y \)-coordinates by \( \frac{5}{2} \):

• For \( J(2, 2) \):

\( x' = \frac{5}{2} \cdot 2 = 5 \), \( y' = \frac{5}{2} \cdot 2 = 5 \) → New \( J'(5, 5) \)

• For \( A(4, 4) \):

\( x' = \frac{5}{2} \cdot 4 = 10 \), \( y' = \frac{5}{2} \cdot 4 = 10 \) → New \( A'(10, 10) \)

• For \( L(8, 2) \):

\( x' = \frac{5}{2} \cdot 8 = 20 \), \( y' = \frac{5}{2} \cdot 2 = 5 \) → New \( L'(20, 5) \)

• For \( M(6, 0) \):

\( x' = \frac{5}{2} \cdot 6 = 15 \), \( y' = \frac{5}{2} \cdot 0 = 0 \) → New \( M'(15, 0) \)

Step 4: Verify with Actual Graph

If the original coordinates differ (e.g., from the grid), adjust the calculation. For example, if a vertex is at \( (x, y) \), multiply both by \( \frac{5}{2} \) to get the dilated coordinates.

Final Answer (Example)

If the original vertex was \( (2, 2) \), the dilated coordinate is \( (5, 5) \). Adjust based on the actual original coordinates from the graph.

(Note: Replace with the correct coordinates from the problem’s graph. For instance, if the original \( J \) is at \( (2, 2) \), the answer for \( J' \) is \( (5, 5) \), etc.)