The circle P has a center at (0,0) and a point on the circle at (0, 4). If it is dilated by a factor of 4, what is the diameter of circle P?
Answer (2)
Determine the radius of the original circle P as the distance from the center (0,0) to the point (0,4), which is 4.
Calculate the radius of the dilated circle by multiplying the original radius by the dilation factor of 4: 4 × 4 = 16 .
Find the diameter of the dilated circle by multiplying the dilated radius by 2: 2 × 16 = 32 .
The distance of the diameter for circle P is 32 .
Explanation
Find the radius of the original circle The circle P has a center at (0,0) and a point on the circle at (0, 4). This means the radius of the circle P is the distance from the center (0,0) to the point (0,4), which is 4.
Calculate the radius of the dilated circle The circle P is dilated by a factor of 4. To find the radius of the dilated circle, we multiply the original radius by the dilation factor: 4 × 4 = 16 . So, the radius of the dilated circle is 16.
Calculate the diameter of the dilated circle The diameter of a circle is twice its radius. Therefore, the diameter of the dilated circle is 2 × 16 = 32 .
State the final answer The distance of the diameter for circle P is 32.
Examples
Imagine you're designing a circular garden and want to enlarge it. If the original garden has a radius of 4 meters and you decide to dilate it by a factor of 4, the new garden will have a radius of 16 meters. Consequently, the diameter of the enlarged garden will be 32 meters, which helps you determine the amount of fencing you'll need to enclose it. This dilation concept is crucial in scaling designs proportionally, whether it's for gardens, blueprints, or even digital graphics.
The radius of the original circle P is 4 units. After dilating the circle by a factor of 4, the new radius becomes 16 units, resulting in a diameter of 32 units. Therefore, the diameter of circle P after dilation is 32 units.
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