Question

A triangle is drawn on a coordinate plane. Dilate the figure with a magnitude of 3 in reference to the origin. What are the coordinates of the new vertices of  A′B′C′ ?(1 point)ResponsesA′(−5,−8), B′(−3,0), and C′(3,0)upper A prime left parenthesis negative 5 comma negative 8 right parenthesis ,  upper B prime left parenthesis negative 3 comma 0 right parenthesis , and  upper C prime left parenthesis 3 comma 0 right parenthesisA′(1,−2), B′(3,6), and C′(9,0)upper A prime left parenthesis 1 comma negative 2 right parenthesis ,  upper B prime left parenthesis 3 comma 6 right parenthesis , and  upper C prime left parenthesis 9 comma 0 right parenthesisA′(−6,−15), B′(0,9), and C′(18,−9)upper A prime left parenthesis negative 6 comma negative 15 right parenthesis ,  upper B prime left parenthesis 0 comma 9 right parenthesis , and  upper C prime left parenthesis 18 comma negative 9 right parenthesisA′(−23,−53), B′(0,1), and C′(2,−1)

Answer 1

To dilate a triangle (or any geometric figure) with a magnitude \( k \) in reference to the origin, you multiply each coordinate of the vertices by \( k \).

Given the dilation magnitude of \( 3 \), the new coordinates \( (x', y') \) of a point \( (x, y) \) after dilation are determined by the following equations:

\[
x' = k \cdot x
\]
\[
y' = k \cdot y
\]

So with \( k = 3 \):

\[
x' = 3x
\]
\[
y' = 3y
\]

You would need to apply these formulas to each of the original vertices of the triangle \( A, B, C \). Since we don’t have the original coordinates provided in your question, I can explain how to compute the new coordinates, but you will need to insert the original coordinates.

Let's say the original coordinates for points \( A, B, C \) are \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \). The new vertices after dilation would be:

- For point \( A' \):
\[
A' = (3x_1, 3y_1)
\]

- For point \( B' \):
\[
B' = (3x_2, 3y_2)
\]

- For point \( C' \):
\[
C' = (3x_3, 3y_3)
\]

Now, if you want to check which of the choices provided corresponds to your original points after applying the dilation, calculate the new coordinates for each option and see which one matches your calculated coordinates.

Since the original points are not presented, make sure to apply the dilation to your specified points and check each response option as a comparison.