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State true or false.

The sum of the three medians of the triangle is always less than the perimeter.

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True

False

answer is A.

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Detailed Solution

Construct a triangle

A B C

D, E, F

are the midpoints of sides

B C, C A

and

A B

respectively as shown in the figure given below.
Question Image

Here, the medians are

A D, B E, C F

.
Perimeter of the triangle

A B C = A B + B C + C A

Analyze that the sum of

2

sides of a triangle is greater than twice the median bisecting the third side, Hence in triangle

A B D

A D

is a median.

A B + A C > 2 A D

......(1)
Similarly,

B C + A C > 2 C F

......(2)

B C + A B > 2 B E

......(3)
Calculate by adding equations (1), (2), and (3).

A B + A C + B C + A C + B C + A B > 2 A D + 2 C F + 2 B E

2 A B + B C + A C > 2 A D + B E + C F

A B + B C + A C > A D + B E + C F

Therefore, the sum of the

3

medians of the triangle is always less than the perimeter.
Hence the statement is true.
Hence option 1 is correct.

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