Question

if M is the midpoint of hypotenuse Ac of right triangle ABC then BM=1\2------

Answer 1

Answer:

BM=1/2 (AC) ....

Step-by-step explanation:

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Answer 2

Concept:

We need the knowledge og geometry and few extra contructions.

A pair of vertically opposite angles are always equal to each other.

Alternate interior andles are always equal.

The sum of the consecutive interior angles is 180°.

CPCT is corresponding parts of Congruent Triangles: If two triangles are congruent, Their corresponding sides are equal. Their corresponding angles are equal.

Given:

M is the midpoint of AC, ABC is right angled triangle at B.

To find:

The value of x in BM=$\frac{1}{2}x$

Solution:

We need we produce BM to E such that BM = ME.  

Now, we know that

AM=MC

BD = DE [By construction]  

And, ∠AMB = ∠CME                [Vertically opposite angles]  

Now using SAS criterion of congruence

∆AMB ≅ ∆CME  

⇒ EC = AB

∠CEM = ∠ABM     ----------(i)

But ∠CED & ∠ABD                  [ alternate interior angles ]  

CE is parallel to AB

⇒ ∠ABC + ∠ECB = 180°          [Consecutive interior angles]  

⇒ 90 + ∠ECB = 180°  

⇒ ∠ECB = 90°  

So now, In ∆ABC & ∆ECB we have, from equation(1)

AB = EC

BC = BC                                    [Common]  

∠ABC = ∠ECB = 90°  

Therefore, by SAS criterion of congruence  

∆ABC ≅ ∆ECB  

⇒ AC = EB                                 [By cpctc]  

⇒  $\frac{1}{2}AC= \frac{1}{2}EB$

⇒ $BM= \frac{1}{2}AC$                            $[\frac{1}{2} EB=BM]$

Therefore, $BM= \frac{1}{2}AC$