a The center of the circle is the midpoint of the segment between the given endpoints.
B
b The area of a circle with radius r can be determined by the formula A=π r^2.
A
a x^2+y^2=400
B
b 1256.64 units^2
Practice makes perfect
a We are told that the sprinkler waters a circular section of the lawn, whose endpoints of the diameter are (- 12,16) and (12,- 16). Let's use this information to make a diagram.
The center of the circle is the midpoint of the diameter. We can find its coordinates using the following formula.
M(x_1+x_2/2,y_1+y_2/2)Here, (x_1,y_2) and (x_2,y_2) are the coordinates of the endpoints of the segment. Let's substitute (x_1,y_1)=( - 12,16) and (x_2,y_2)=( 12,- 16) into the formula and simplify.
We conclude that the center of the circle is at (0,0). Now, let's find the radius of the circle. In order to do this we will use the Distance Formula.
AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Here, (x_1,y_1) and (x_2,y_2) are the coordinates of the endpoints A and B. We want to find the radius, which is the distance from the center to every point on the circumference of a circle. Let's find the distance from M to K by substituting (x_1,y_1) with ( 0,0) and (x_2,y_2) with ( 12,- 16) into the formula.
Now that we know both the center and the radius of the circle, we can write its equation. To do this, let's recall that if a circle has the center at (h,k) and the radius of r, its equation can be written the following way.
(x-h)^2+(y-k)^2=r^2
Substituting ( h, k) with ( 0, 0) and r with 20, we get the following equation.
(x- 0)^2&+(y- 0)^2=20^2
&⇓
x^2&+y^2=400
This equation represents the boundary of the sprinkler area.
b The area of a circle with the radius of r can be determined using the following formula.
A=π r^2
From Part A, we know that the radius of the circular lawn the sprinkler waters is 20. Let's substitute r with 20 into the formula and solve it for A.
