Question

Point B(-2, 4) lies on a circle centered at A(1, 3). Write a paragraph proof to determine whether C(4, 2) also lies on the circle.

Answer 1

I am given that the center of a circle is at A(1, 3) and that point B(-2, 4) lies on the circle. Applying the distance formula to A and B, I get the following:

Since B lies on the circle, this length is the length of the radius of the circle. Applying the distance formula to A and C(4, 2), I get the following:

Thus, the distance to C from the center A is equal to the length of the radius of the circle. Any point whose distance from the center is equal to the length of the radius lies on the circle. Therefore, point C lies on the circle.

Explanation:

Answer 2

we know that

if point B (-2,4) lies on a circle centered at A (1,3)

then

distance AB is equal to the radius of the circle

so

step 1

find the distance AB

dAB=√[(y2-y1)²+(x2-x1)²]

dAB=√[(4-3)²+(1+2)²]------> dAB=√[1²+3²]------> √10 units

step 2

Find the equation of a circle

(x-h)²+(y-k)²=r²--------> (x-1)²+(y-3)²=10

step 3

Any point on the circle has to satisfy the equation

(x-1)²+(y-3)² = 10

We replace x and y with the given point C (4,2).

If we replace x with 4 and y with 2

we get

(4-1)² + (2-3)² = 9+1

10=10

Because both sides of the equation are equal this means that the point

C (4,2) is on the circle.

Any point that is not on the circle won't satisfy the equation.

Only points on the circle will give us the result of 10 = 10