A triangle has corners at #(7 ,5 )#, #(2 ,3 )#, and #(1 ,4 )#. What is the area of the triangle's circumscribed circle?

Answer 1

The area of the circumscribed circle is:

#Area = 2146/196pi#

The standard Cartesian form of the equation of a circle is:

#(x - h)^2 + (y - k)^2 = r^2" [1]"#
where #(x, y)# is any point on the circle, #(h, k)# is the center point, and r is the radius.
When I do a problem of this type, I shift all 3 points so that one point is then origin, #(0, 0)#, because this simplifies the problem and it does not change the area of this circle:
#(2,3) to (0,0)#
#(1,4) to (-1,1)#
#(7,5) to (5, 2)#

Use equation [1] and the new points to write 3 equation:

#(0 - h)^2 + (0 - k)^2 = r^2" [2]"# #(-1 - h)^2 + (1 - k)^2 = r^2" [3]"# #(5 - h)^2 + (2 - k)^2 = r^2" [4]"#

Equation [2] simplifies into:

#h^2 + k^2 = r^2" [5]"#

Substitute the left side of equation [5] into the right sides of equations [3] and [4]:

#(-1 - h)^2 + (1 - k)^2 = h^2 + k^2" [6]"# #(5 - h)^2 + (2 - k)^2 = h^2 + k^2" [7]"#

Expand the squares:

#1 + 2h + h^2 + 1 - 2k + k^2 = h^2 + k^2" [8]"# #25 - 10h + h^2 + 4 - 4k + k^2 = h^2 + k^2" [9]"#
The #h^2 and k^2# terms sum to zero:
#1 + 2h + 1 - 2k = 0" [10]"# #25 - 10h + 4 - 4k = 0" [11]"#

Collect the constant terms into a single term on the right:

#2h - 2k = -2" [10]"# #-10h -4k = -29" [11]"#

Multiply equation [10] by -2 and add to equation [11]:

#-14h = -25#
#h = 25/14#

Substitute #25/14 for h in equation [10] and the solve for k:

#2(25/14) - 2k = -2#
#k = 1 + 25/14#
#k = 39/14#

Substitute the values for h and k into equation [5]

#r^2 = (25/14)^2 + (39/14)^2#
#r^2 = 2146/196#

The area of a circle is:

#Area = pir^2#

The area of the circumscribed circle is:

#Area = 2146/196pi#
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Answer 2

To find the area of the triangle's circumscribed circle, you can use the formula for the area of a circle, which is π times the square of the circle's radius. The radius of the circumscribed circle of a triangle is equal to half the length of the triangle's longest side. So, first, you need to find the lengths of the sides of the triangle using the given coordinates. Then, identify the longest side and calculate its length. Finally, divide the length of the longest side by 2 to find the radius of the circumscribed circle. Once you have the radius, you can use it to calculate the area of the circumscribed circle using the formula for the area of a circle.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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