Question

polygon wxz has vertices w(1, 0), x(-2, 1), x(-3, -2), z(2, -3). 11. draw and label polygon wxz. 12. draw and label the image of wxyz with a center of dilation at (0,0) and a dilation of 2. label the image wxyz. 13. explain how you could determine the coordinates of the dilated image wxyz without graphing.

Explanation:

Question 11: Draw and label polygon \( WXYZ \)

Step 1: Identify the coordinates of each vertex

• \( W(1, 0) \): Move 1 unit to the right on the x - axis and 0 units on the y - axis.
• \( X(-2, 1) \): Move 2 units to the left on the x - axis and 1 unit up on the y - axis.
• \( Y(-3, -2) \): Move 3 units to the left on the x - axis and 2 units down on the y - axis.
• \( Z(2, -3) \): Move 2 units to the right on the x - axis and 3 units down on the y - axis.

Step 2: Plot the points on the coordinate plane

Using the coordinate grid, mark the points \( W(1,0) \), \( X(-2,1) \), \( Y(-3,-2) \), and \( Z(2,-3) \) with dots.

Step 3: Connect the points in order

Draw line segments to connect \( W \) to \( X \), \( X \) to \( Y \), \( Y \) to \( Z \), and \( Z \) to \( W \) to form the polygon \( WXYZ \). Then label each vertex with its corresponding letter.

Question 12: Draw and label the image of \( WXYZ \) with a center of dilation at \( (0,0) \) and a dilation of 2

Step 1: Recall the rule for dilation about the origin

The rule for a dilation with scale factor \( k \) about the origin \((0,0)\) is \((x,y)\to(kx,ky)\). Here, \( k = 2 \).

Step 2: Apply the dilation rule to each vertex

• For \( W(1,0) \): \( W'=(2\times1,2\times0)=(2,0) \)
• For \( X(-2,1) \): \( X'=(2\times(-2),2\times1)=(-4,2) \)
• For \( Y(-3,-2) \): \( Y'=(2\times(-3),2\times(-2))=(-6,-4) \)
• For \( Z(2,-3) \): \( Z'=(2\times2,2\times(-3))=(4,-6) \)

Step 3: Plot the dilated points and connect them

Plot the points \( W'(2,0) \), \( X'(-4,2) \), \( Y'(-6,-4) \), and \( Z'(4,-6) \) on the coordinate plane. Then connect them in order ( \( W' \) to \( X' \), \( X' \) to \( Y' \), \( Y' \) to \( Z' \), \( Z' \) to \( W' \)) and label the image as \( W'X'Y'Z' \).

Question 13: Explain how to determine the coordinates of the dilated image \( W'X'Y'Z' \) without graphing

Step 1: State the dilation rule

When a figure is dilated with a center of dilation at the origin \((0,0)\) and a scale factor \( k \), the coordinates of each vertex \((x,y)\) of the original figure are transformed to \((kx,ky)\) in the dilated figure.

Step 2: Apply the rule to each vertex of \( WXYZ \)

• For vertex \( W(1,0) \): Multiply the x - coordinate (1) and y - coordinate (0) by the scale factor 2. So \( W'=(2\times1,2\times0)=(2,0) \).
• For vertex \( X(-2,1) \): Multiply the x - coordinate (-2) and y - coordinate (1) by 2. So \( X'=(2\times(-2),2\times1)=(-4,2) \).
• For vertex \( Y(-3,-2) \): Multiply the x - coordinate (-3) and y - coordinate (-2) by 2. So \( Y'=(2\times(-3),2\times(-2))=(-6,-4) \).
• For vertex \( Z(2,-3) \): Multiply the x - coordinate (2) and y - coordinate (-3) by 2. So \( Z'=(2\times2,2\times(-3))=(4,-6) \).

By following this rule of multiplying each coordinate of the original vertices by the scale factor of the dilation, we can find the coordinates of the dilated image without graphing.

Answer:

Question 11: Draw and label polygon \( WXYZ \)

Step 1: Identify the coordinates of each vertex

• \( W(1, 0) \): Move 1 unit to the right on the x - axis and 0 units on the y - axis.
• \( X(-2, 1) \): Move 2 units to the left on the x - axis and 1 unit up on the y - axis.
• \( Y(-3, -2) \): Move 3 units to the left on the x - axis and 2 units down on the y - axis.
• \( Z(2, -3) \): Move 2 units to the right on the x - axis and 3 units down on the y - axis.

Step 2: Plot the points on the coordinate plane

Using the coordinate grid, mark the points \( W(1,0) \), \( X(-2,1) \), \( Y(-3,-2) \), and \( Z(2,-3) \) with dots.

Step 3: Connect the points in order

Draw line segments to connect \( W \) to \( X \), \( X \) to \( Y \), \( Y \) to \( Z \), and \( Z \) to \( W \) to form the polygon \( WXYZ \). Then label each vertex with its corresponding letter.

Question 12: Draw and label the image of \( WXYZ \) with a center of dilation at \( (0,0) \) and a dilation of 2

Step 1: Recall the rule for dilation about the origin

The rule for a dilation with scale factor \( k \) about the origin \((0,0)\) is \((x,y)\to(kx,ky)\). Here, \( k = 2 \).

Step 2: Apply the dilation rule to each vertex

• For \( W(1,0) \): \( W'=(2\times1,2\times0)=(2,0) \)
• For \( X(-2,1) \): \( X'=(2\times(-2),2\times1)=(-4,2) \)
• For \( Y(-3,-2) \): \( Y'=(2\times(-3),2\times(-2))=(-6,-4) \)
• For \( Z(2,-3) \): \( Z'=(2\times2,2\times(-3))=(4,-6) \)

Step 3: Plot the dilated points and connect them

Plot the points \( W'(2,0) \), \( X'(-4,2) \), \( Y'(-6,-4) \), and \( Z'(4,-6) \) on the coordinate plane. Then connect them in order ( \( W' \) to \( X' \), \( X' \) to \( Y' \), \( Y' \) to \( Z' \), \( Z' \) to \( W' \)) and label the image as \( W'X'Y'Z' \).

Question 13: Explain how to determine the coordinates of the dilated image \( W'X'Y'Z' \) without graphing

Step 1: State the dilation rule

When a figure is dilated with a center of dilation at the origin \((0,0)\) and a scale factor \( k \), the coordinates of each vertex \((x,y)\) of the original figure are transformed to \((kx,ky)\) in the dilated figure.

Step 2: Apply the rule to each vertex of \( WXYZ \)

• For vertex \( W(1,0) \): Multiply the x - coordinate (1) and y - coordinate (0) by the scale factor 2. So \( W'=(2\times1,2\times0)=(2,0) \).
• For vertex \( X(-2,1) \): Multiply the x - coordinate (-2) and y - coordinate (1) by 2. So \( X'=(2\times(-2),2\times1)=(-4,2) \).
• For vertex \( Y(-3,-2) \): Multiply the x - coordinate (-3) and y - coordinate (-2) by 2. So \( Y'=(2\times(-3),2\times(-2))=(-6,-4) \).
• For vertex \( Z(2,-3) \): Multiply the x - coordinate (2) and y - coordinate (-3) by 2. So \( Z'=(2\times2,2\times(-3))=(4,-6) \).

By following this rule of multiplying each coordinate of the original vertices by the scale factor of the dilation, we can find the coordinates of the dilated image without graphing.