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Prove that the Perimeter of a Triangle is Greater than the Sum of Its Altitudes. - Mathematics

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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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 12 Congruent Triangles
Exercise 12.6 | Q 6 | Page 81
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