Given: two opposites vertices of square are (1,2) and (5,8)

To find: opposite’s vertices of a square and equation of sides.

Explanation:

Let A (1, 2) be the vertex of square ABCD and BD be the diagonal that is along the line 8x − 15y = 0

Equation of the given line is, 8x – 15y = 0

⇒ - 15y = - 8x

Comparing this equation with y = mx + c

We get, m

So, the slope of BD will be .
Here, we have to find the equations of sides AB and AD.

We know that the equations of two lines passing through a point x1,y1 and making an angle α with the line whose slope is m.

Here,

m, x₁ = 1, y₁ = 2, α = 45∘

So, the equations of the required sides are

⇒ 23x – 7y – 9 = 0 and 7x + 23y – 53 = 0

Hence, equation of sides is 23x – 7y – 9 = 0 and 7x + 23y – 53 = 0