Question

determine the length of the diameter, in simplest radical form, of ⊙m when given the center of the circle and a point on the edge of the circle as shown. show your algebraic thinking to earn full credit. m(3,4) n(5,4)

Explanation:

Step1: Find the radius (distance between M and N)

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 \(M(3,4)\) and \(N(5,4)\), \(x_1 = 3\), \(y_1 = 4\), \(x_2 = 5\), \(y_2 = 4\).
\[

$$\begin{align*} r&=\sqrt{(5 - 3)^2+(4 - 4)^2}\\ &=\sqrt{2^2+0^2}\\ &=\sqrt{4}\\ & = 2 \end{align*}$$

\]

Step2: Find the diameter (twice the radius)

The diameter \(d\) of a circle is \(d = 2r\). Since \(r = 2\), then \(d=2\times2 = 4\).

Answer:

The length of the diameter is \(4\).