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Q9 Circle T passes through the origin and the point U(3,-4). What is the circumference of Circle T?

Q99\newlineCircle T \mathrm{T} passes through the origin and the point U(3,−4) U(3,-4) . What is the circumference of Circle T?

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Q. Q99\newlineCircle T \mathrm{T} passes through the origin and the point U(3,−4) U(3,-4) . What is the circumference of Circle T?
  1. Find Radius of Circle T: To find the circumference of Circle T, we first need to determine the radius of the circle. The radius can be found by calculating the distance between the center of the circle, which is at the origin (0,0)(0,0), and the point U(3,−4)U(3,-4) that lies on the circle.
  2. Calculate Distance Formula: The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane is given by the formula: distance=(x2−x1)2+(y2−y1)2\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. For the origin (0,0)(0,0) and point U(3,−4)(3,-4), the distance (radius) is: r=(3−0)2+(−4−0)2r = \sqrt{(3 - 0)^2 + (-4 - 0)^2}.
  3. Calculate Radius: Plugging the values into the distance formula, we get: r=(32+(−4)2)=(9+16)=25r = \sqrt{(3^2 + (-4)^2)} = \sqrt{(9 + 16)} = \sqrt{25}.
  4. Find Circumference Formula: After calculating the square root of 2525, we find that the radius r=5r = 5 units.
  5. Substitute Radius: Now that we have the radius, we can find the circumference of Circle T using the formula for the circumference of a circle, which is C=2πrC = 2\pi r, where rr is the radius.
  6. Calculate Circumference: Substituting the radius into the circumference formula, we get: C=2π(5)=10πC = 2\pi(5) = 10\pi.

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