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)
Responses

A′(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 parenthesis

A′(−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 parenthesis

A′(−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 parenthesis

A′(−23,−53), B′(0,1), and C′(2,−1)
upper A prime left parenthesis negative Start Fraction 2 over 3 End Fraction comma negative Start Fraction 5 over 3 End Fraction right parenthesis , upper B prime left parenthesis 0 comma 1 right parenthesis , and upper C prime left parenthesis 2 comma negative 1 right parenthesis

Answer 1

To dilate a triangle from the origin by a factor of 3, you multiply the coordinates of each vertex by 3.

For vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) of the triangle:

- The new coordinates \( A' \) will be \( (3x_1, 3y_1) \)
- The new coordinates \( B' \) will be \( (3x_2, 3y_2) \)
- The new coordinates \( C' \) will be \( (3x_3, 3y_3) \)

Assuming the original coordinates of vertices \( A \), \( B \), and \( C \) were not provided, the answer cannot be calculated without that information. However, if we do take the already provided responses:

1. Response 1: A′(1,−2), B′(3,6), and C′(9,0)
2. Response 2: A′(−6,−15), B′(0,9), and C′(18,−9)
3. Response 3: A′(−5,−8), B′(−3,0), and C′(3,0)
4. Response 4: A′(−23,−53), B′(0,1), and C′(2,−1)

Let's check if any of these responses could result from an original triangle coordinates.

To make sure we correctly identify, let's denote:

1. If the original coordinates were \( A( \frac{1}{3}, -\frac{2}{3}), B(1,2), C(3,0) \), thus producing \( A′(1,−2), B′(3,6), and C′(9,0) \).

Since we need more details on original coordinates to examine, we would select \( A'(−6,−15) \), \( B′(0,9) \), and \( C′(18,−9) \).

Therefore the only viable coordinates could be:

- The original coordinates of \(A', B', C'\) leading us towards our potential option yielding and confirming calculations by exploring the listed vertices:

After checking through the responses carefully, without additional information, we would assume therefore we found:

**Final Response:**
A′(1,−2), B′(3,6), C′(9,0) will be correct under related coordinates definition aligning with dilation approach too.