Given the $$ riangle A B C $$ with the points $$ A=(-1,3) B=(2,4) C=(4,7) $$ and it's dilation, triangle $$ A^{\prime} B^{\prime} C^{\prime} $$, with points $$ A^{\prime}=(-3,9) B^{\prime}=(6,12) C^{\prime}=(12,21) $$ what is the scale factor? Select one: a. $$ \frac{1}{3} $$ b. 2 c. $$ \frac{1}{2} $$ d. 3

Question

Given the $$ 	riangle A B C $$ with the points $$ A=(-1,3) B=(2,4) C=(4,7) $$ and it's dilation, triangle $$ A^{\prime} B^{\prime} C^{\prime} $$, with points $$ A^{\prime}=(-3,9) B^{\prime}=(6,12) C^{\prime}=(12,21) $$ what is the scale factor? Select one: a. $$ \frac{1}{3} $$ b. 2 c. $$ \frac{1}{2} $$ d. 3

UpStudy AI · Accepted Answer

We start by checking the transformation for vertex $$ A $$. The dilation is given by:

$$
A = (-1, 3) \quad \text{and} \quad A' = (-3, 9)
$$

Assuming the dilation is centered at the origin, each coordinate is multiplied by the scale factor $$ k $$. Thus, we have:

$$
-1 \times k = -3 \quad \text{and} \quad 3 \times k = 9
$$

Solving the first equation:

$$
k = \frac{-3}{-1} = 3
$$

Now, we verify this using the second coordinate:

$$
3 \times 3 = 9
$$

Both coordinates give the same scale factor $$ k = 3 $$.

Next, we verify with vertex $$ B $$:

$$
B = (2, 4) \quad \text{and} \quad B' = (6, 12)
$$

Check the $$ x $$-coordinate:

$$
2 \times k = 6 \quad \Rightarrow \quad k = \frac{6}{2} = 3
$$

And the $$ y $$-coordinate:

$$
4 \times 3 = 12
$$

Finally, for vertex $$ C $$:

$$
C = (4, 7) \quad \text{and} \quad C' = (12, 21)
$$

Check the $$ x $$-coordinate:

$$
4 \times k = 12 \quad \Rightarrow \quad k = \frac{12}{4} = 3
$$

And the $$ y $$-coordinate:

$$
7 \times 3 = 21
$$

Since all vertices yield a scale factor $$ k = 3 $$, the dilation is confirmed to have a scale factor of 3.

Thus, the answer is:

d. $$ 3 $$
The scale factor is 3.

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