Given $$ riangle A B C $$ with vertices at $$ -A(1,2), B(3,7) $$, and $$ C(10,-2) $$, and it's dilation, $$ riangle A^{\prime} B^{\prime} C $$, with vertices at $$ -A^{\prime}(6,12), B^{\prime}(18,42) $$, and $$ C^{\prime}(60,-12) $$. What is the scale factor? Select one: a. $$ \frac{1}{3} $$ b. $$ \frac{1}{2} $$ c. 3 d. 6

Question

Given $$ 	riangle A B C $$ with vertices at $$ -A(1,2), B(3,7) $$, and $$ C(10,-2) $$, and it's dilation, $$ 	riangle A^{\prime} B^{\prime} C $$, with vertices at $$ -A^{\prime}(6,12), B^{\prime}(18,42) $$, and $$ C^{\prime}(60,-12) $$. What is the scale factor? Select one: a. $$ \frac{1}{3} $$ b. $$ \frac{1}{2} $$ c. 3 d. 6

UpStudy AI · Accepted Answer

Let the dilation have a center at the origin. The dilation sends each point $$ (x, y) $$ to $$ (kx, ky) $$. We are given:

$$
A(1,2) \rightarrow A^{\prime}(6,12)
$$
$$
B(3,7) \rightarrow B^{\prime}(18,42)
$$
$$
C(10,-2) \rightarrow C^{\prime}(60,-12)
$$

Check $$ A $$ first:
$$
k \cdot 1 = 6 \quad \Rightarrow \quad k = 6,
$$
$$
k \cdot 2 = 12 \quad \Rightarrow \quad k = 6.
$$

Check $$ B $$:
$$
k \cdot 3 = 18 \quad \Rightarrow \quad k = 6,
$$
$$
k \cdot 7 = 42 \quad \Rightarrow \quad k = 6.
$$

Check $$ C $$:
$$
k \cdot 10 = 60 \quad \Rightarrow \quad k = 6,
$$
$$
k \cdot (-2) = -12 \quad \Rightarrow \quad k = 6.
$$

Since the scale factor is consistently $$ k = 6 $$, the answer is:

d. $$6$$
The scale factor is 6.

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