A triangle has corners at #(5 ,8 )#, #(2 ,7 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(5  ,8   )#, #(2   ,7   )#, and #(7   ,3  )#.  What is the area of the triangle's circumscribed circle?

Theodore Abad · Accepted Answer

#~~ 32.3#

Let the center of the circum-circle be #(h,k)# and its radius be #r#. Then, the equation of the circle is

#(x-h)^2+(y-k)^2 = r^2#

Since this circle must pass through the three vertices we have

#(5-h)^2+(8-k)^2 = (2-h)^2+(7-k)^2 = (7-h)^2+(3-k)^2 = r^2#

From #(5-h)^2+(8-k)^2 = (2-h)^2+(7-k)^2# we get

#25-10h+h^2 +64-16k+k^2 = 4-4h+k^2+49-14k+k^2#

so that

#color(red)(6h + 2k = 36)#

Similarly, #(2-h)^2+(7-k)^2 = (7-h)^2+(3-k)^2# gives us

#4-4h+h^2 +49-14k+k^2 = 49-14h+h^2+9-6k+k^2 # which simplifies to

#color(red)(10h-8k = 5)#

Solving the two equations simultaneously gives

# h = 149/34,quad k = 165/34#

This leads to #r^2 ~~ 1.29 #

so that the area of the circumscribed circle is #pi r^2 ~~ 32.3#

Claire Akin · Answer

undefined To find the area of the triangle's circumscribed circle, you first need to find the circumradius, which is the radius of the circle that circumscribes the triangle. Then, you can use the formula for the area of a circle (π * radius^2).

1. Calculate the lengths of the sides of the triangle using the distance formula:

Side 1: Distance between (5, 8) and (2, 7)
   Side 2: Distance between (5, 8) and (7, 3)
   Side 3: Distance between (2, 7) and (7, 3)

2. Use Heron's formula to find the area of the triangle using the side lengths.

3. Use the formula for the circumradius of a triangle, which is given by:
   
   $ R = \frac{{abc}}{{4A}} $

where $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle, and $ A $ is the area of the triangle.

4. Once you find the circumradius, use the formula for the area of a circle to find the area of the circumscribed circle.

Let's go through these steps:

1. Calculate the side lengths:
   Side 1: $ \sqrt{(5-2)^2 + (8-7)^2} = \sqrt{9} = 3 $
   Side 2: $ \sqrt{(5-7)^2 + (8-3)^2} = \sqrt{29} $
   Side 3: $ \sqrt{(2-7)^2 + (7-3)^2} = \sqrt{25} = 5 $

2. Use Heron's formula to find the area of the triangle:
   $ s = \frac{{3 + \sqrt{29} + 5}}{2} = \frac{{8 + \sqrt{29}}{2} $
   $ A = \sqrt{s(s-3)(s-\sqrt{29})(s-5)} $

3. Calculate the area using Heron's formula:
   $ A = \sqrt{(\frac{{8 + \sqrt{29}}}{2})((\frac{{8 + \sqrt{29}}}{2})-3)((\frac{{8 + \sqrt{29}}}{2})-\sqrt{29})((\frac{{8 + \sqrt{29}}}{2})-5)} $
   
4. Calculate the circumradius:
   $ R = \frac{{3 \cdot \sqrt{29} \cdot 5}}{{4 \cdot A}} $

5. Once you find $ R $, use the formula for the area of a circle:
   $ Area = \pi R^2 $