If in acute-angled triangle ABC, AL, BM, and CN are the three altitudes of triangle ABC, then which of the following statements will be true?

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 27 Jul 2023 Shift 2)

Answer 1

Given

AL, BM, and CN are the three altitudes of triangle ABC

Concept used

The side opposite to the greater angle is a greater

Calculation

In the above diagram,

∠ALB = 90

∠ABL < 90

Now side opposite to ∠ALB is AB and the side opposite to ∠ABL is AL

Since the side opposite to the greater angle is a greater

AB > AL ..(1)

Similarly in triangle BMC

BC > BM ..(2)

Similarly in triangle ANC

CA > CN ..(3)

Adding all three:

AL + BM + CN < AB + BC + CA

The answer is AL + BM + CN < AB + BC + CA

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