A triangle has corners at #(7 ,8 )#, #(2 ,3 )#, and #(1 ,4 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(7  ,8   )#, #(2   ,3   )#, and #(1   ,4  )#.  What is the area of the triangle's circumscribed circle?

Sophia Alfred · Accepted Answer

Area of the circle = 1.2752

the triangle is a right-angled triangle. The side lengths are #sqrt2,sqrt50 and sqrt52, respectively# Triangle area (A) = #1/2xxsqrt2xxsqrt50=sqrt100/2=10/2=5# triangle perimeter (P) = #sqrt2+sqrt50+sqrt52=15.696# inradius (R)= #2A/P=2xx5/15.696=0.6371# Area of the circle = #pixxR^2=pixx0.6371^2=1.2752#

Claire Akin · Answer

undefined To find the area of the circumscribed circle of a triangle, we need to find the circumradius, which is the radius of the circle passing through all three vertices of the triangle.

First, calculate the lengths of the sides of the triangle using the distance formula. Then, use Heron's formula to find the area of the triangle. Finally, use the formula $ A = \frac{{abc}}{{4R}} $, where $ A $ is the area of the triangle, $ a $, $ b $, and $ c $ are the lengths of its sides, and $ R $ is the circumradius.

After finding the area of the triangle and the lengths of its sides, solve for the circumradius using the given formula. Once you have the circumradius, you can find the area of the circumscribed circle using the formula for the area of a circle ($ A = \pi R^2 $).

Given the coordinates of the triangle's vertices, you can follow these steps to calculate the area of the circumscribed circle.