In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If ∠BAC=45° and ∠ABC=θ, then AD/BE equals

Question

UPSC Quizroom · Accepted Answer

Let a, b, c be the side lengths opposite A, B, C.
Let Δ be the area of triangle ABC.

Given:
Angle A = 45°
Angle B = θ

Step 1: Use altitude formula for area

Area = (1/2) * a * AD
Area = (1/2) * b * BE

So,
a * AD = b * BE

Therefore,
AD / BE = b / a (1)

Step 2: Use Sine Rule

a / sin(A) = b / sin(B)

So,
a = k * sin(A)
b = k * sin(B)
for some constant k.

Thus,
b / a = sin(B) / sin(A) (2)
Step 3: Substitute values

A = 45°
B = θ

So,
b / a = sin(θ) / sin(45°)

Now,
sin(45°) = √2 / 2

Therefore,
b / a = sin(θ) / (√2 / 2)
b / a = √2 * sin(θ)

Step 4: From (1),

AD / BE = b / a = √2 * sin(θ)

Final Answer:
√2 sinθ

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