Answer:
CD = 6.385 units
Step-by-step explanation:
Given triangle ABC with right angle at C.
And AB = AD + 6 .
Now, consider the triangle ABC.
⇒ cos(∠BAC) = [tex]\frac{AC}{AB}[/tex] (cosФ = adj/hyp)
cos(20) = [tex]\frac{AC}{AB}[/tex] .
0.9397 = [tex]\frac{AD+CD}{AD + 6}[/tex]
(since AB = AD + 6 and AC = AD + CD)
⇒ 0.9397 AD + 5.6382 = AD + CD
⇒ CD = 0.0603 AD + 5.6382. →→→→→ (1)
⇒ sin(∠BAC) = [tex]\frac{BC}{AB}[/tex] (sinФ = opp/hyp)
sin(20) = [tex]\frac{BC}{AB}[/tex].
⇒ BC = AB sin(20) . →→→→→(2)
Now, consider the triangle BCD,
sin(∠BDC) = [tex]\frac{BC}{CD}[/tex]
⇒ sin(80) = [tex]\frac{BC}{CD}[/tex]
CD = [tex]\frac{BC}{sin(80)}[/tex]
From (2), CD = [tex]\frac{AB sin(20)}{sin(80)}[/tex] .
⇒ CD = AB (0.3473)
⇒ CD = (AD + 6) (0.3473)
⇒ CD = 0.3473 AD + 2.0838 →→→→→→(3)
Now, (1) →→ CD = 0.0603 AD + 5.6382
(3) →→ CD = 0.3473 AD + 2.0838
⇒ 0.0603 AD + 5.6382 = 0.3473 AD + 2.0838
0.287 AD = 3.5544.
⇒ AD = 12.3847
⇒ From (1), CD = 0.0603(12.3847) + 5.6382
⇒ CD = 6.385 units
