Answer in Brief

Prove that the perimeter of a triangle is greater than the sum of its altitudes.

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#### Solution

We have to prove that the perimeter of a triangle is greater than the sum of its altitude.

In ΔABC

AD⊥ BC , BE ⊥ AC , CF⊥AB

We have to prove

AB + BC + CD > AD + BE + CF

Since AD⊥ BC

So AB > AD and AC > AD

By adding AB + AC > AD + AD, we have

AB + AC > 2AD ........(1)

Now consider BE ⊥ AC then

BC > BE, and BA > BE

Now by adding BC + BA > 2BE .......(2)

Again consider CF⊥AB

AC > CF, and BC > CF

By adding AC + BC > 2CF ...........(3)

Adding (1), (2) and (3), we get

2(AB + BC + CA)>2 (AD + BE + CF)

⇒ AB + BC + CA > AD + BE + CF

Hence the perimeter of a triangle is greater than the sum of all its altitude.

Concept: Criteria for Congruence of Triangles

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