Question

In AABC, seg AM is the median. AB + AC = 410 and BC = 12. What is the
length of seg AM?
​

Answer 1

the answer is AM= 12units

Answer 2

Therefore the length of the line segment AM is 13 units.

Given:

In ΔABC, segment AM is the median.

AB² + AC² = 410 and BC = 12

To Find:

The length of the line segment AM.

Solution:

The given question can be easily solved as shown below.

Given that,

In ΔABC,

⇒ Line segment AM is the median.

⇒ AB² + AC² = 410

⇒ BC = 12

From the figure attached, BM = CM = 6 units

ΔAMC and ΔAMB are right-angled triangles.

In ΔAMC, by applying Pythagoras theorem,

⇒ AC² = MC² + AM²

⇒ AC² = 6² + AM²________(i.)

In ΔAMB, by applying Pythagoras theorem,

⇒ AB² = MB² + AM²

⇒ AB² = 6² + AM²_________(ii.)

Adding equations (i.) and (ii.), we get,

⇒ AC² + AB² = ( 6² + AM² ) + ( 6² + AM² )

⇒ AC² + AB² = 36 + 36 + AM² + AM²

⇒ AC² + AB² = 72 + 2AM²

⇒ 410 = 72 + 2AM²    ( ∵ AC² + AB² = 410 )

⇒ 2AM² = 410 - 72 = 338

⇒ AM² = 338 / 2 = 169

⇒ AM = 13

Therefore the length of the line segment AM is 13 units.

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