Question

If (–5, 3) and (5, 3) are two vertices of an equilateral triangle, then find the coordinates of third vertex, given that origin lies inside the triangle. `("Take"  sqrt(3) = 1.7)`

Answer 1

Given: Two vertices A(–5, 3) and B(5, 3) of an equilateral triangle; origin (0,0) lies inside the triangle.

Step-wise calculation:

1. Midpoint of AB is M = `((-5 + 5)/2, (3 + 3)/2) = (0, 3)`. 

By symmetry the third vertex C lies on the perpendicular bisector x = 0, so let C = (0, y).

2. Side length AB = Distance between A and B = 10, so AB² = 100.

3. AC² = (0 – (–5))² + (y – 3)²

= 25 + (y – 3)²

For an equilateral triangle AC = AB,

So 25 + (y – 3)² = 100

⇒ (y – 3)² = 75 

⇒ y – 3 = `±sqrt(75)`

⇒ y – 3 = `±5sqrt(3)`

4. Hence `y = 3 ± 5sqrt(3)`.

Using the instruction `sqrt(3) = 1.7`

Compute `5sqrt(3) = 5 xx 1.7 = 8.5`

So y = 3 + 8.5 = 11.5 or y = 3 – 8.5 = –5.5.

5. Since the origin must lie inside the triangle, choose the vertex with y between the base (y = 3) and the lower vertex so that (0, 0) is inside. The correct choice is y = –5.5.

The third vertex is C = (0, –5.5).