Answer:
Step-by-step explanation:
CD = 4sin30 = 4(1/2) = 2
∠BDC = 180 - 90 - 60 = 30°
BDcos30 = CD = 2
BD = 2/cos30 = 2/ (½√3) = 4/√3 = (4/3)√3
another way to do it would be
∠BDC = 180 - 90 - 60 = 30°
∠ADC = 180 - 90 - 30 = 60°
therefore
∠BDA = 60 - 30 = 30°
We can see the upper triangle is an isosceles triangle where lengths AB = BD and the angle between them is 180 - 30 - 30 = 120°
Using law of Sines
4/sin120 = BD/sin30
BD = 4sin30/sin120 = 4(½)/(½√3) = 4/√3 = (4/3)√3
or we could use the law of Cosines
c² = a² + b² - 2abcosC
BD² = AB² + 4² - 2(AB)(4)cos30
BD = AB = x
x² = x² + 4² - 2(x)(4)cos30
0 = 4² - 2(x)(4)cos30
4² = 2(x)(4)cos30
x = 4²/(2(4cos30)) = 2/cos30 = 2/ (½√3) = 4/√3 = (4/3)√3
