A diameter of a circle has endpoints P(-7, -4) and Q (3,2).
a. Find the center of the circle (hint: use midpoint formula).
b. Find the radius. If your answer is not and integer, express in radical form. (hint: use distance formula).
(You should get radius = √34 for your answer).
c. Write an equation for the circle.
(You should get equation: (x+2)² + (y+1)² = 34 for your answer).
Show work.
(This is 10th grade math, please don't make it advanced, shorten the steps).
a. Midpoint formula for center:
x-coordinate: (-7+3)/2 = -2
y-coordinate: (-4+2)/2 = -1
Center of circle is (-2,-1).
b. Distance formula for radius:
√[(3-(-7))² + (2-(-4))²]
= √[10² + 6²]
= √(100 + 36)
= √136
Simplified, the radius is √34.
c. Using center and radius:
Equation of circle:
(x+2)² + (y+1)² = (√34)²
(x+2)² + (y+1)² = 34
a. To find the center of the circle, we need to find the midpoint of the line segment PQ.
The midpoint formula is:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the given endpoints:
Midpoint = ((-7 + 3)/2, (-4 + 2)/2)
= (-4/2, -2/2)
= (-2, -1)
Therefore, the center of the circle is (-2, -1).
b. To find the radius, we need to find the distance between the center of the circle and either of the endpoints.
The distance formula is:
Distance = √((x2 - x1)² + (y2 - y1)²)
Using the center (-2, -1) and one of the endpoints, let's use P (-7, -4):
Distance = √((-7 - (-2))² + (-4 - (-1))²)
= √((-7 + 2)² + (-4 + 1)²)
= √((-5)² + (-3)²)
= √(25 + 9)
= √34
Therefore, the radius of the circle is √34.
c. To write the equation for the circle, we use the formula:
Standard Form: (x - h)² + (y - k)² = r²
Where (h, k) is the center of the circle and r is the radius.
Substituting the values:
(x - (-2))² + (y - (-1))² = (√34)²
(x + 2)² + (y + 1)² = 34
Therefore, the equation for the circle is (x + 2)² + (y + 1)² = 34.