Question

given two circles that are similar on a coordinate plan. the first circle \\(a\\) has a center at \\((3, 2)\\) and a radius of 2. translate circle a by the rule \\((x-6, y-4)\\), then dilate it by a factor of 2. what are the coordinates of the new circles center and its radius?

center \\((3, -1)\\) radius 4
center \\((-3, 2)\\) radius 4
center \\((-3, -2)\\) radius 4
center \\((-3, 1)\\) radius 2

Explanation:

Identify the initial properties of circle A

The first circle \(A\) has:

• Center: \((3, 2)\)
• Radius: \(r = 2\)

Apply the translation rule

Translate the center \((3, 2)\) using the rule \((x - 6, y - 4)\):

$$ x' = 3 - 6 = -3 $$

$$ y' = 2 - 4 = -2 $$

The translated center is \((-3, -2)\).
Translation does not change the radius, so \(r' = 2\).

Apply the dilation factor

Dilate the translated circle by a factor of 2.
Dilation centered at the origin scales the coordinates of the center and the radius:

• New Center: \((-3 \times 2, -2 \times 2) = (-6, -4)\) (if centered at the origin).
• However, standard geometric transformations of a shape on a coordinate plane usually dilate the shape with respect to its own center or a specified center. If no center of dilation is specified, we must check the options to see which convention is used.
• Let's check the options:
• Option 1: Center \((3, -1)\) radius 4
• Option 2: Center \((-3, 2)\) radius 4
• Option 3: Center \((-3, -2)\) radius 4
• Option 4: Center \((-3, 1)\) radius 2
• Looking at Option 3: Center \((-3, -2)\) radius 4. This corresponds to the translated center \((-3, -2)\) and a dilated radius of \(2 \times 2 = 4\). This indicates the dilation is performed with respect to the circle's own center, meaning only the radius is scaled by 2, while the center remains at the translated position \((-3, -2)\).

Confirm the final coordinates and radius

• Final Center: \((-3, -2)\)
• Final Radius: \(2 \times 2 = 4\)

This matches the third option.

Answer:

• Center \((3, -1)\) radius 4
• Center \((-3, 2)\) radius 4</mcq-correct>
• Center \((-3, -2)\) radius 4 (Correct answer)

<mcq-option>Center \((-3, 1)\) radius 2