Question

find the area of the triangle formed by the lines y-x=0,x+y=0 and x-k=0.

Answer 1

Let equations of line
A : y - x = 0
B : x + y = 0
C : x = k , where k is constant .

first of all solve equations A and B
y - x = 0
x + y = 0
we get x = 0, and y = 0
hence, one point is ( 0, 0) of traingle .

again, solve equations B and C
x + y = 0
x = k
then, we get x = k And y = -k
so , ( k, - k) is 2nd point of traingle .

similarly solve equations C and A
y - x = 0 , x = k
we get ( k, k ) is third point of traingle.

now, we have three points
( 0, 0) , ( k, - k) and ( k, k)

we know, if three points of traingle are given then we have to use for finding area of traingle is 1/2 [x₁(y₂-y₃) + x₂(y₃-y₁) +x₃(y₁-y₂) ]

so, area of traingle = 1/2 [ 0( - k - k ) + k( k - 0) + k( 0 + k) ]
= 1/2 [ 0 + k² + k² ]
= 2k²/2
= k²

hence, area of traingle = k²