Question

Question 1. Find the center and radius of the circunscribed circle.

Answer 1

In order to find the center of the circunscribed circle, we can use the midpoint theorem because the center point is in the middle of any two vertices

that is, if we take points (9,23) and (8,16) the midpoint C is given as

`C=((8+9)/(2),(16+23)/(2))`

which gives

`C=(8.5,19.5)`

So the center of the circle is the point (8.5,19.5)

On the other hand, the radius is equal to the distance from any vertex to the center. If we take the vertex (8,16), we get

`r=\sqrt[]{(8.5-8)^2+(19.5-16)^2}`

which gives

`\begin{gathered} r=\sqrt[]{0.5^2+3.5^2} \\ r=\sqrt[]{0.25+12.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}`

so, the radius measure 3.54 units.

Now, lets prove that the answer are correct. In order to do that, we can choose the other vertices and apply the same procedure as above.

So the vertices are (5,20) and (12,19). Again, the center is the midpoint between these points and is given as

`\begin{gathered} C=((5+12)/(2),(20+19)/(2)) \\ C=((17)/(2),(39)/(2)) \\ C=(8.5,19.5) \end{gathered}`

which is the same center as above.

Now, the distance from the center to vertec (5,20) is

`\begin{gathered} r=\sqrt[]{(8.5-5)^2+(20-19.5)^2} \\ r=\sqrt[]{3.5^2+0.5^2} \\ r=\sqrt[]{12.25+0.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}`

which is the same radius obtained above. Then, the answers are correct.