I came across a question today.
Two mutually perpendicular straight lines through the origin forms an isosceles triangle with the line $2x + y = 5$. Then the area of the triangle is ?
I know that it can be solved by finding the length ($l$) of perpendicular from origin to the line and then the length of the hypotenuse and then using $\text{Area }= \frac{ab}{2} = l^2$. And the area came out to be $5$ (which is a correct answer).
But I tried a different approach. I took the perpendicular straight lines as $x$-axis and $y$-axis and found the coordinates of the line on the axes using intercept form of line. They came out to be $(\frac{5}{2}, 0)$ and $(0,5)$.
Now as these points works as base and altitude of the triangle. So $\text{Area}=\left(\frac{1}{2}\right)\left(\frac{5}{2}\right)\left(5\right)=\frac{25}{4}$
What is the mistake in the second solution?
Thanks to @imranfat for answering in the comments.
It is because the $x$-axis (given by $y=0$), $y$-axis (given by $x=0$), and $2x+y=5$ do not form a isosceles triangle when it is clearly given in the question that the triangle should be isosceles.