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Two equal sides of an isosceles triangle...

Two equal sides of an isosceles triangle are given by `7x-y+3=0` and `x+y=3`, and its third side passes through the point `(1,-10)`. Find the equation of the third side.

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Since triangle is isosceles, the third side is equally inclined to the lines 7x-y+3 = 0 and x+y-3=0.
Hence, the third side is parallel to angle bisectors of the given lines.

The equation of the two bisectors of given lines are
`(7x-y+3)/(sqrt(50))= +-(x+y-3)/(sqrt(2))`
` "or " 3x+y-3=0 " " (1)`
` "and " x-3y+9=0 " " (2)`
Equation of line through (1,-10) and parallel to (1) is 3x+y+7 = 0.
Equation of line through (1,-10) and parallel to (2) is x-3y-31=0.
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