SOLUTION: The lines 4y=7x+21, 4y=31-3x and x-2y=-3 intersect at point A, B and C. Find the co-ordinates of these points and find the area of triangle ABC

Algebra ->  Graphs -> SOLUTION: The lines 4y=7x+21, 4y=31-3x and x-2y=-3 intersect at point A, B and C. Find the co-ordinates of these points and find the area of triangle ABC      Log On


   

Question 1054654: The lines 4y=7x+21, 4y=31-3x and x-2y=-3 intersect at point A, B and C. Find the co-ordinates of these points and find the area of triangle ABC
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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Use the distance formula to find the distance between the points.
a=sqrt%28%28-3-1%29%5E2%2B%280-7%29%5E2%29=sqrt%2816%2B49%29=sqrt%2865%29
b=sqrt%28%28-3-5%29%5E2%2B%280-4%29%5E2%29=sqrt%2864%2B16%29=sqrt%2880%29
c=sqrt%28%281-5%29%5E2%2B%287-4%29%5E2%29=sqrt%2816%2B9%29=sqrt%2825%29=5
Use Heron's formula,
s=%28a%2Bb%2Bc%29%2F2=%28sqrt%2865%29%2Bsqrt%2880%29%2B5%29%2F2
A=sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29


s-c=sqrt%2865%29%2F2%2Bsqrt%2880%29%2F2%2B5%2F2-5=sqrt%2865%29%2F2%2Bsqrt%2880%29%2F2-5%2F2
Let
X=sqrt%2865%29%2F2
Y=sqrt%2880%29%2F2
Z=5%2F2
s=X%2BY%2BZ
s-a=Y%2BZ-X
s-b=X%2BZ-Y
s-c=X%2BY-Z
So then,
A=sqrt%28%28X%2BY%2BZ%29%28Y%2BZ-X%29%28X%2BZ-Y%29%28X%2BY-Z%29%29
A=sqrt%28X%5E2%28-X%5E2%2B2Y%5E2%2B2Z%5E2%29-Y%5E4%2BZ%5E2%282Y%5E2-Z%5E2%29%29
Substituting,

A=sqrt%286400%2F16%29
A=sqrt%28400%29
A=20