Question

ABC is an equilateral triangle, BD ⊥ AC, BD = `5sqrt3`. Find:

1. AD
2. the perimeter of ΔABC.

Answer 1

Given:

BD = `5sqrt3`

△ABC is equilateral.

So, altitude BD ⊥ AC and bisects AC.

AD = DC = `a/2`, where a = side of the equilateral triangle.

(i) Using right triangle ABD:

AB² = AD² + BD²

Substitute

AB = a

AD = `a/2`

BD = `5sqrt3`

= a² − `(a/2)^2 + (5sqrt3)`

= `a^2 = a^2/4 + 75`

= `a^2 − a^2/4 + 75`

= `(3a^2)/4 = 75`

a² = 100 

a = 10

AD = `a/2` 

AD = `10/2`

= 5 cm

(ii) Perimeter of △ABC = 3a:

= 3 × 10

= 30 cm