Question

how are the coordinates of a dilated point calculated?
a. divide each coordinate by the scale factor
b. add the scale factor to each coordinate
c. subtract the center of dilation from the coordinates
d. multiply each coordinate by the scale factor

dilate point ( a(3, -4) ) by a scale factor of ( k = 3 ) with the center of dilation at the origin. what are the coordinates of ( a )?
a. ( (9, -12) )
b. ( (1, -1.33) )
c. ( (9, -9) )
d. ( (3, -3) )

if a triangle with vertices at ( (2, 3) ), ( (4, 5) ), and ( (6, 7) ) is dilated with a scale factor of 3 using the origin, what are the new coordinates of the vertex ( (2, 3) )?
a. ( (4, 5) )
b. ( (1, 1.5) )
c. ( (2, 3) )
d. ( (6, 9) )

Explanation:

First Question: How are the coordinates of a dilated point calculated?

Dilation in coordinate geometry involves resizing a point (or figure) relative to a center of dilation. The rule for dilation with a scale factor \( k \) (when the center is the origin, or generally, for each coordinate) is to multiply each coordinate of the original point by the scale factor. Option a is incorrect (division is for reduction in some cases but not the standard rule), option b is incorrect (adding scale factor is not dilation), option c is incorrect (subtracting center is part of a different transformation or adjusting for center, not the dilation of coordinates themselves). Option d matches the dilation rule.

Step1: Recall the dilation rule

For a point \( (x, y) \) dilated with scale factor \( k \) from the origin, the new coordinates are \( (k \cdot x, k \cdot y) \).

Step2: Apply the rule to point \( A(3, -4) \)

Multiply the \( x \)-coordinate: \( 3 \times 3 = 9 \)
Multiply the \( y \)-coordinate: \( -4 \times 3 = -12 \)
So the coordinates of \( A' \) are \( (9, -12) \).

Step1: Recall the dilation rule

For a point \( (x, y) \) dilated with scale factor \( k \) from the origin, the new coordinates are \( (k \cdot x, k \cdot y) \).

Step2: Apply the rule to \( (2, 3) \)

Multiply the \( x \)-coordinate: \( 2 \times 3 = 6 \)
Multiply the \( y \)-coordinate: \( 3 \times 3 = 9 \)
So the new coordinates are \( (6, 9) \).

Answer:

d. Multiply each coordinate by the scale factor

Second Question: Dilate point \( A(3, -4) \) by a scale factor of \( k = 3 \) with the center of dilation at the origin. What are the coordinates of \( A' \)?