A triangle has corners at #(5 ,8 )#, #(3 ,9 )#, and #(4 ,7 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(5  ,8   )#, #(3   ,9   )#, and #(4   ,7  )#.  What is the area of the triangle's circumscribed circle?

Evelyn Arboleda · Accepted Answer

the area of the circle inscribed = 0.8160

the triangle is isoceles, side lengths = #sqrt5, sqrt5 and sqrt2# triangle area (A) = #1/2xxsqrt2xx3/sqrt2=3/2# triangle perimeter (P) = #sqrt5+sqrt5+sqrt2=2sqrt5+sqrt2=5.8863# inradius (r) =#(2A)/P# =# 2xx1.5/5.8863=0.50965# area of the circle = #pixxr^2 = pixx(0.50965)^2# # =0.8160#

Mason Adams · Answer

undefined To find the area of the circumscribed circle of a triangle, you can use the formula:

$$ Area = \frac{abc}{4R} $$

Where $ a, b, $ and $ c $ are the side lengths of the triangle, and $ R $ is the radius of the circumscribed circle. First, calculate the side lengths using the given coordinates:

$$ a = \sqrt{(5-3)^2 + (8-9)^2} $$
$$ b = \sqrt{(5-4)^2 + (8-7)^2} $$
$$ c = \sqrt{(4-3)^2 + (7-9)^2} $$

Next, find the semiperimeter $ s $ of the triangle:

$$ s = \frac{a + b + c}{2} $$

Then, use Heron's formula to find the area $ A $ of the triangle:

$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$

Once you have the area of the triangle, you can find the radius $ R $ of the circumscribed circle using the formula:

$$ R = \frac{abc}{4A} $$

Finally, calculate the area of the circumscribed circle using the formula for the area of a circle:

$$ Area = \pi R^2 $$

Substitute the calculated value of $ R $ into this formula to get the area of the circumscribed circle.