In the given figure if AD ⊥ BC, AC = 26 units, CD = 10 units, BC = 42 units, ∠DAC = x and ∠B = y then the value of \(\rm \frac{6}{\cos x}-\frac{5}{\cos y}+8\tan y\) is:
25/4 units
Given:
AD ⊥ BC ; ∠D = 90°
AC = 26 units; CD = 10 units; BC = 42 units
∠DAC = x and ∠B = y
Formula used:
Pythagorean theorem:
H2 = P2 + B2
Cos θ = B/H ; tan θ = P/B
Where, H = hypotenuse; P = perpendicular; B = Base
Calculation:
In △DAC
⇒ AC2 = AD2 + CD2
⇒ 262 = AD2 + 102
⇒ 676 = AD2 + 100
⇒ AD = √(676 - 100) = √576 = 24 units
In △ADB
BD = (BC - CD) = (42 - 10) = 32 units
⇒ AB2 = AD2 + BD2
⇒ AB2 = 242 + 322
⇒ AB2 = 576 + 1024
⇒ AB = √1600 = 40 units
From the question:
\(\rm \frac{6}{\cos x}-\frac{5}{\cos y}+8\tan y\)
⇒ 6/(AD/AC) - 5/(BD/AB) + 8 × (AD/BD)
⇒ 6/(24/26) - 5/(32/40) + 8 × (24/32)
⇒ (6 × 26/24) - (5 × 40/32) + 8 × (24 /32 )
⇒ 6.5 - 6.25 + 6
⇒ 6.25 = 25/4 units
∴ The correct answer is 25/4 units.
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