Answer:
CD = 5.196 cm
Area = 31.177 sq. cm.
Step-by-step explanation:
See the attached diagram.
Given that ∠ ACB = 90° in Δ ABC.
Now, CD ⊥ AB and ∠ CDB = ∠ CDA = 90°
Given that ∠ ACD = 60° and BC = 6 cm.
We have to find the length of CD and the area of Δ ABC.
Now, ∠ CAD = 90° - ∠ ACD = 90° - 60° = 30°
Again, ∠ CBD = 90° - ∠ CAD = 90° - 30° = 60°.
Now, from Δ BCD, [tex]\sin 60 = \frac{CD}{BC} = \frac{CD}{6}[/tex]
{Since Δ BCD is a right triangle and ∠ CDB = 90°}
⇒ [tex]CD = 6 \times \sin 60 = 5.196[/tex] cm. (Answer)
Now, from Δ ACD, [tex]\sin 30 = \frac{CD}{AC} = \frac{5.169}{AC}[/tex]
{Since Δ ACD is a right triangle and ∠ ADC = 90°}
⇒ [tex]AC = \frac{5.196}{\sin 30} = 10.392[/tex] cm
So, the area of Δ ABC = [tex]\frac{1}{2} \times BC \times AC = \frac{10.392 \times 6}{2} = 31.177[/tex] sq. cm. (Answer)
