Question

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in the xy-plane above, points a and b are on the circle with center o. what is length of \\(\overline{ab}\\)?
(image: a circle with center o at the origin, point a(3,4) on the circle, point b on the x-axis on the circle, with segments oa, ab, and ob drawn.)

Explanation:

Step1: Find the radius OA (and OB, since they are radii)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For point \(A(3,4)\) and center \(O(0,0)\), the length of \(OA\) is \(\sqrt{(3 - 0)^2+(4 - 0)^2}=\sqrt{9 + 16}=\sqrt{25} = 5\). Since \(OB\) is also a radius of the circle (center \(O\)), \(OB=OA = 5\), so the coordinates of \(B\) are \((5,0)\) (because it's on the x - axis and the distance from \(O\) is 5).

Step2: Calculate the length of \(AB\)

Now we use the distance formula between \(A(3,4)\) and \(B(5,0)\). The distance \(AB=\sqrt{(5 - 3)^2+(0 - 4)^2}=\sqrt{2^2+(- 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Answer:

\(2\sqrt{5}\)