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?

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SSC CGL 2023 Tier-I Official Paper (Held On: 27 Jul 2023 Shift 2)
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  1. AL + BM + CN = AB + BC + CA 
  2. AL + BM = AB + BC 
  3. AL + BM + CN > AB + BC + CA 
  4. AL + BM + CN < AB + BC + CA 

Answer (Detailed Solution Below)

Option 4 : AL + BM + CN < AB + BC + CA 
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Detailed Solution

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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

F3 Savita SSC 28-12-23 D4

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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