Question

Without distance formula, show that the points A (5,8), B (4,4), C (0,5) and D (1,9) form a rhombus.

Answer 1

We have to prove that ABCD is a rhombus

Slope of AC = `("y"_2 - "y"_1)/("x"_2 - "x"_1) = (5 - 8)/(0 - 5) = (-3)/(-5) = 3/5`

Slope of BD = `("y"_2 - "y"_1)/("x"_2 - "x"_1) = (9- 4)/(1 - 4)  = 5/(-3)`

Thus,Slope of AC x slope of BD = -1

So, the diagnols AC and BC are perpendicular to each other.

Hence,ABCD is a rhombus.