Given: two side of an isosceles triangle are 7x – y + 3 = 0 and x + y – 3 = 0 and its third side passes through the point (1, -10)

To find: third side of isosceles triangle

Explanation:

Let ABC be the isosceles triangle, where 7x − y + 3 = 0 and x + y − 3 = 0 represent the sides AB and AC, respectively.
Let AB = BC

Diagram:

∵ AB = BC

∴ tan B = tan C

Here,

Slope of AB = 7

Slope of AC = − 1

Let m be the slope of BC.

Then,

Taking the positive sign, we get:

m² – 8m + 7 = 7m² + 8m + 1

Now, taking the negative sign, we get:

(m – 7) (m – 1) = - (7m + 1)(m + 1)

⇒ m² – 8m + 7 = - 7m² – 8m – 1

⇒ m² = - 1 (not possible)

Equations of the third side is

y + 10 = - 3(x – 1) and

⇒ 3x + y + 7 = 0 and x – 3y – 31 = 0

Hence, third side of isosceles triangle is 3x + y + 7 = 0 and x – 3y – 31 = 0