State whether the statement is true or false.

The perimeter of a triangle is greater than the sum of its three altitudes.

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True

False

answer is A.

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Detailed Solution

Given the statement, the perimeter of a triangle is greater than the sum of its three altitudes.
Draw a figure relating the given statement.
Question Image

In the figure, AD

⊥

BC.
In a right angled triangle, the hypotenuse is the largest side.
AB > AD and AC > AD. [since,

Δ

ABD and

Δ

ADC are right angles with AB and AC as hypotenuse respectively]
We know that the sum of any two triangle sides is more than twice the median with respect to the third side.
So, AB + AC > 2AD …(1)
Similarly, BE

⊥

AC.
BA > BE and BC > BE.

\Rightarrow

BA + BC > 2BE …(2)
And similarly, CF

⊥

AB.
CA > CF and CB > CF

\Rightarrow

CA + CB > 2CF …(3)
Now add (1), (2), (3).

\Rightarrow

AB + AC + BA + BC + CA + CB = 2AD + 2BE + 2CF

\Rightarrow

AB + BC + CA > AD + BE + CF
So we can say that the perimeter of the triangle is greater than the sum of its altitudes.
So, the given statement is true and the correct option is 1.

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