Question

In an acute triangle ABC, AL perpendicular BC. Prove that AB > BL and AC > CL. Hence deduce the
result AB + AC > BC.​

Answer 1

Given :  In an acute triangle ABC, AL perpendicular BC.

To find : Prove that AB > BL and AC > CL  also AB + AC > BC.​

Solution:

In an acute triangle ABC, AL perpendicular BC

=> Δ ABL is right angle triangle

AB is hypotenuse

hypotenuse is largest side of right angle triangle

( or use the fact that side opposite to largest angle is largest and 90° is largest angle )

=> AB > BL

Similarly

=> Δ ACL is right angle triangle

AC is hypotenuse

hypotenuse is largest side of right angle triangle

( or use the fact that side opposite to largest angle is largest and 90° is largest angle )

=> AC > CL

AB + BC  > BL + CL

=> AB + BC > BC    ( also sum of two sides of triangle is greater than third side )

QED

Proved

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