From the above figure find length of AC. Given that angle ABD =30 , angle ADC =45 BD = 40m
Answers
Answer:
AC = 40(√3 + √2) m
OR
AC = (40√3 + 40√2) m
Step-by-step explanation:
We are given that,
∠ABD = 30°
∠ADC = 45°
BD = 40 m
We know that,
Tan B = AC/BC
Tan B = Tan 30°
Tan 30° = 1/√3
also, BC = BD + CD = CD + 40 m
Thus,
(1/√3) = AC/(CD + 40)
AC/(1/√3) = CD + 40
AC√3 - 40 = CD ---- 1
Now,
Tan D = AC/CD
Tan D = Tan 45
Tan 45 = 1/√2
1/√2 = AC/CD
Thus,
CD = AC/(1/√2)
CD = AC√2 ------ 2
From eq.1 and eq.2 we get,
AC√2 = AC√3 - 40
40 = AC√3 - AC√2
AC(√3 - √2) = 40
Thus,
AC = 40/(√3 - √2)
Rationalizing the denominator,
AC = 40(√3 + √2)/(√3 - √2)(√3 + √2)
Using (a + b)(a - b) = a² - b²
AC = 40(√3 + √2)/(√3)² - (√2)²
AC = 40(√3 + √2)/3 - 2
Thus,
AC = 40(√3 + √2) m
OR
AC = (40√3 + 40√2) m
Hope it helped and you understood it........All the best
Solution :-
Let us assume that, length of DC is equal to x cm .
So,
→ BC = BD + DC = (40 + x) m
now , in right angled ∆ABC ,
→ tan 30° = AC/BC
→ 1/√3 = AC/(40 + x)
→ AC = {(40 + x)/√3} m -------- Eqn.(1)
also, in right angled ∆ADC ,
→ tan 45° = AC/DC
→ 1 = AC/x
→ AC = x m ------------- Eqn.(2)
from Eqn.(1) and Eqn.(2),
→ x = (40 + x)/√3
→ √3x = 40 + x
→ √3x - x = 40
→ x(√3 - 1) = 40
→ x = 40(√3 - 1)
rationalizing the denominator,
→ x = {40(√3 - 1)} * {(√3 + 1)/(√3 + 1)}
→ x = 40(√3 + 1)/(3 - 1)
→ x = 20(√3 + 1)
from Eqn.(2) we get,
→ AC = x
→ AC = 20(√3 + 1) m (Ans.)
Hence, Length of AC is equal to 20(√3 + 1) m .
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