Answer:
To find the ratio that relates YZ to XY, we can use the distance formula.
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We need to find the lengths of segments YZ and XY, and then determine the ratio of these lengths.
First, let's find the length of segment YZ:
\[ d_{YZ} = \sqrt{(6 - 4)^2 + (2 - 0)^2} \]
\[ d_{YZ} = \sqrt{2^2 + 2^2} \]
\[ d_{YZ} = \sqrt{8} \]
Now, let's find the length of segment XY:
\[ d_{XY} = \sqrt{(6 - (-6))^2 + (2 - (-2))^2} \]
\[ d_{XY} = \sqrt{(6 + 6)^2 + (2 + 2)^2} \]
\[ d_{XY} = \sqrt{12^2 + 4^2} \]
\[ d_{XY} = \sqrt{144 + 16} \]
\[ d_{XY} = \sqrt{160} \]
Now, we can find the ratio of \( d_{YZ} \) to \( d_{XY} \):
\[ \text{Ratio} = \frac{d_{YZ}}{d_{XY}} = \frac{\sqrt{8}}{\sqrt{160}} \]
To simplify the ratio, we can take the square root of the numerator and denominator:
\[ \text{Ratio} = \frac{\sqrt{8}}{\sqrt{160}} = \frac{2\sqrt{2}}{4\sqrt{10}} = \frac{\sqrt{2}}{2\sqrt{10}} \]
Now, let's rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{10} \):
\[ \text{Ratio} = \frac{\sqrt{2} \cdot \sqrt{10}}{2\sqrt{10} \cdot \sqrt{10}} = \frac{\sqrt{20}}{20} = \frac{2\sqrt{5}}{20} = \frac{\sqrt{5}}{10} \]
So, the ratio of YZ to XY is \( \frac{\sqrt{5}}{10} \). This can be simplified further by dividing both the numerator and the denominator by \( \sqrt{5} \):
\[ \text{Ratio} = \frac{\sqrt{5}}{10} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{1}{2} \]
So, the ratio of YZ to XY is 1:2, which corresponds to option A.
Step-by-step explanation:
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