if-any-angle-of-equilateral-triangle-is-1-2-and-any-one-side-is-x-3y-5-0-then-equation-of-the-other-two-sides-are-

Question Number 8608 by tawakalitu last updated on 17/Oct/16

if any angle of equilateral triangle is

(- 1, 2) and any one side is x - \sqrt{3 y} + 5 = 0

then equation of the other two sides are ? ?

Answered by sandy_suhendra last updated on 18/Oct/16

Answered by sandy_suhendra last updated on 18/Oct/16

let △ ABC is the equilateral triangle

the equation of B \overset{}{C} \Rightarrow x - \sqrt{3} y + 5 = 0

so the gradient of BC = m_{BC} = \tan ∠ BPQ = \frac{1}{3} \sqrt{3} \Rightarrow ∠ BPQ = 30 °

∠ ACB = 60 ° \Rightarrow ∠ PCA = 180 ° - 60 ° = 120 °

∠ CRQ = ∠ BPQ + ∠ PCA = 30 ° + 120 ° = 150 °

(AC intercepts X axis at R)

the gradient of AC = \tan ∠ CRQ = \tan 150 ° = - \frac{1}{3} \sqrt{3}

the equation of AC \Rightarrow y = m_{AC} (x - x_{A}) + y_{A}

y = - \frac{1}{3} \sqrt{3} (x + 1) + 2

∠ BQP = 180 ° - ∠ BPQ - ∠ CBA = 180 ° - 30 ° - 60 ° = 90 ° = ∠ BQX

the gradient of AB = \tan ∠ BQX = \tan 90 ° = \infty

the equation of AB \Rightarrow x = - 1

Commented by tawakalitu last updated on 18/Oct/16

Thank you for your help .