Question

The hypotenuse of a right angled triangle has its ends at the points (1, 3) and (−4, 1). Find the equation of the legs (perpendicular sides) of the triangle that are parallel to the axes.

Answer 1

Let triangle ABC be a right angled triangle whose hypotenuse is AB. The coordinates of A and B are (1, 3) and (−4, 1) respectively.

Let the slope of BC be m.

AC ⊥ BC

∴ Slope of AC = `-1/"m"`

Equation of line BC,

y – y₁ = m(x – x₁)

y – 1 = m(x + 4)

or mx – y + 4m + 1 = 0      .......(i)

Equation of line AC

y – 3 = `- 1/"m" ("x" - 1)`

or my – 3m = – x + 1

or x + my – 3m – 1 = 0        ........(ii)

The equation of both these lines can be found from the given value of m. If side BC is parallel to x-axis, then m = 0

Equation of BC, y – 1 = 0

or y = 1

∴ AC is parallel to y-axis and it goes through A(1, 3). Hence, the equation of AC is x = 1

Hence, the equations of BC and AC are y = 1 and x = 1.