Question

A triangle ABC with point D on segment CA . AB=4,BD=3,AD=5, angle DBC=30 .If DC=K(4 root 3 +3 , compute the rational number K.

Answer 1

Considering the triangle ABD, AD² = AB² + BD² [5² = 4² + 3²]
So it is a right triangle with <ABD = 90 deg.
<DBC = 30 deg (Given)
So, <ABC = 120 deg.

Let <BAC = A

Applying angle sum property to triangle ABC,
<ACB = 180 - (120 + A) = 60 - A


iii) Applying sine law to the triangle ABC,
AC/sin(<ABC) = AB/sin(<ACB)

==> AC/sin(120) = 4/sin(60 - A)

So, AC = 4*sin(120)/sin(60 - A) ------------- (1)

sin(60 - A) = sin(60)*cos(A) - cos(60)*sin(A)
[From triangle ABD, cos(A) = AB/AD = 4/5 and sin(A) = BD/AD = 3/5; as well cos(60) = 1/2 and sin(60) = sin(120) = √3/2]

Substituting these in (1) above,
AC = (4*√3/2)/{(√3/2)*(4/5) - (1/2)*(3/5)}

Simplifying, AC = 20√3/(4√3 - 3)

Rationalizing the denominator and simplifying,
AC = 80/13 + 20√3/13 -------- (2)

From the given data, AC = AD + DC = 5 + 4k√3 + 3k
==> AC = (3k + 5) + 4k√3 ----- (3)

Equating (2) & (3): (3k + 5) + 4k√3 = 80/13 + 20√3/13

Equating either rational parts or irrational parts and solving, k = 5/13

Thus value of k = 5/13