Question

1. the points (-5, -8) and (5, 2) are located on a circle, as shown in the graph. the line segment connecting the two points represents a diameter of the circle.

reference hw 2.8, #s 4 and 7
a. use the distance formula to calculate the length of the circles diameter. if necessary, express the answer in simplified radical form and then round to two decimal places. (show your work.)
exact answer: ____________
approx. answer: ____________
b. what is the circles radius?
c. use the mid - point formula to find the coordinates of the circles center point. (show your work.)
center: ____________

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(- 5,-8)$ and $(x_2,y_2)=(5,2)$.

Step2: Substitute values into formula

$d=\sqrt{(5-(-5))^2+(2 - (-8))^2}=\sqrt{(5 + 5)^2+(2 + 8)^2}=\sqrt{(10)^2+(10)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}$

Step3: Approximate the value

$\sqrt{2}\approx1.414$, so $10\sqrt{2}\approx10\times1.414 = 14.14$

Step4: Find the radius

The radius $r$ of a circle is half of the diameter $d$. Since $d = 10\sqrt{2}$, then $r=\frac{d}{2}=5\sqrt{2}$

Step5: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$

Step6: Substitute values for mid - point

For points $(-5,-8)$ and $(5,2)$, the mid - point (center of the circle) is $(\frac{-5 + 5}{2},\frac{-8+2}{2})=(0,-3)$

Answer:

a. exact answer: $10\sqrt{2}$
approx. answer: $14.14$
b. $5\sqrt{2}$
c. $(0,-3)$