straight lines

for Q.1, solve the equations two at a time to get the respective coordinates of the vertices of the triangle....then use heron's formula.

for Q.2 , make an equation of the line joining the given points, find its slope, and use the relation for slopes of 2 perpendicular lines i.e. m₁m₂= -1.

1)need not use heron's formula,from equations it can be easily established that the lines y - x=0 and x +y=0 are perpendicular and then we can apply --area=1/2 base x height.

also both these lines intersect at origin and base=height = k√2 (using trignometry)

so area is k²....

2)22/9

Question : 1

To find the area of triangle OAB ...

It simple because OAB is a right angled triangle..

Area = ( AB) ( OM) / 2

Area = ( 2K) ( K) / 2 = K^2 and.

Question : 2

.the line 7x-9y-19=0 is perpendicular to the line through the points(x,3)
and (4,1), find the value of x.

Two lines having slope m1 and m2 are perpendicular ...then m1.m2 = -1

Now slope of the line passing through (x,3) and (4,1) ...is (3-1)/(x-4) = 2/(x-4)

Slope of the line 7x-9y = 19 is 7/9

SO 2/(x-4) * 7/9 = - 1

or 2/(x-4) = -9/7

14 = -9x + 36

or 9x = 22

x = 22/9 ans