P(-6, 1) and Q(6, 6) are the two ends of the diameter of a given circle. Calculate the radius.
6.5 units
13.0 units
3.5 units
7.0 units
Explanation
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To find the radius of the circle when given the coordinates of two endpoints of its diameter, you can use the distance formula.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Given the points \(P(-6, 1)\) and \(Q(6, 6)\), we can use these coordinates to find the distance between them, which is equal to the diameter of the circle. Then, we divide this diameter by 2 to get the radius.
So, substituting the coordinates into the formula:
\[ d = \sqrt{(6 - (-6))^2 + (6 - 1)^2} \]
\[ d = \sqrt{(6 + 6)^2 + (6 - 1)^2} \]
\[ d = \sqrt{12^2 + 5^2} \]
\[ d = \sqrt{144 + 25} \]
\[ d = \sqrt{169} \]
\[ d = 13 \]
Now, since the diameter is 13 units, the radius of the circle is half of that:
\[ \text{Radius} = \frac{13}{2} = 6.5 \]
So, the radius of the circle is 6.5 units.
