Answer: y = 12
Explanation:
Since line BD is the angle bisector of angle ABC, it means that BD divides angle ABC into two equal halves. Therefore,
3x - 1 = 34 - 2x
3x + 2 = 34 + 1
5x = 35
x = 35/5 = 7
Angle ABD = 3x - 1 = 3 × 7 - 1 = 20 degrees
Angle CBD = 34 - 2x = 34 - 2 × 7 = 20 degrees.
Considering right triangle ABD and CBD, the common hypotenuse is AD.
The opposite side of triangle ABD is 3y + 6.
Applying sine rule,
Sin20 = (3y + 6)/AB
AB = (3y + 6)/Sin20
The opposite side of triangle CBD is 5y - 18.
Applying sine rule,
Sin20 = (5y - 18)/AB
AB = (5y - 18)/Sin20
Therefore, AB = AB
(3y + 6)/Sin20 = (5y - 18)/Sin20
Cross multiplying,
20(3y + 6) = 20(5y - 18)
Dividing through by 20
3y + 6 = 5y - 18
3y - 5y = - 18 - 6
- 2y = - 24
y = - 24/-2 = 12