Question

In ΔABC, m∠B=90 and AC=10. The length of the median BM = ......,select a proper option (a), (b), (c) or (d) from given options so that the statement becomes correct.
(a) 5
(b) 5√2
(c) 6
(d) 8

Answer 1

in ∆ABC , m∠B=90°
means, ∆ABC is right angled triangle.
In Δ ABC, use the theorem of Apolloneous,
we know that,
For BM to be the median, we have,
AB² + AC² = 2(BM² + CM²)
Also, as Δ ABC is right angled at B,
⇒ AB² + AC² = AC² = 10² = 100
⇒ 100 = 2 (BM² + 5²)
[ CM is the midpoint of AC = 10]
⇒ 50 = BM² + 25
⇒ BM = 5
∴ Option (a) is correct.

Answer 2

Given :

In ∆ABC , <B = 90° ,

AM is a median .

AC = 10 ,

AM = AC/2

i ) In ∆ABC , <B = 90°

By Phythogarian theorem ,

AC² = AB² + BC² ---( 1 )

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By Appolonius Theorem :

The sum of the squares of two

sides of a triangle is equal to

twice the square on half the

third side plus twice the square

on the median which bisects the

third side .

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Here ,

AB² + BC² = 2( AM² + BM² )

=> AC² = 2 [ ( AC/2 )² + BM² ] { from (1 ) ]

=> 10² = 2 [ 5² + BM² ]

=> 100/2 = 25 + BM²

=> 50 - 25 = BM²

=> 25 = BM²

=> BM = √25

=> BM = 5

Option ( a ) is correct.

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