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In triangle $ABC,$ let $M$ be the midpoint of $\overline{AB}.$ (a) Prove that if $CM = \frac{1}{2} AB,$ then $\angle ACB = 90^\circ.$ (b) Prove that if $\angle ACB = 90^\circ,$ then $CM = \frac{1}{2} AB.$

 Jul 7, 2019

Best Answer 

 #1
avatar+8853 
+5

(a)  https://web2.0calc.com/questions/math-halp-plz#r1

 

(b)

 

 

 

 

Let     AB  =  c     (because it is the side across from angle C)

and    AC  =  b     (because it is the side across from angle B)

and    BC  =  a     (because it is the side across from angle A)

 

Draw a height from  M  which meets side  AC  at point  D

 

m∠BAC  =  m∠MAD     because they are the same angle

m∠ACB  =  m∠ADM     because they are both right angles

 

So by AA similarity,  △ABC ~ △AMD

 

And we know   AM  =  c / 2   because  M  is the midpoint of  AB

 

So the scale factor from  △ABC  to  △AMD  is  1/2   And so...

 

AM  =  c / 2

AD  =  b / 2

DM  =  a / 2

 

Then by SAS congruence we can determine that  △ADM  ≅  △CDM   and so...

 

CM  =  AM

CM  =  c / 2

CM  =  (1/2)(AB)

 Jul 7, 2019
 #1
avatar+8853 
+5
Best Answer

(a)  https://web2.0calc.com/questions/math-halp-plz#r1

 

(b)

 

 

 

 

Let     AB  =  c     (because it is the side across from angle C)

and    AC  =  b     (because it is the side across from angle B)

and    BC  =  a     (because it is the side across from angle A)

 

Draw a height from  M  which meets side  AC  at point  D

 

m∠BAC  =  m∠MAD     because they are the same angle

m∠ACB  =  m∠ADM     because they are both right angles

 

So by AA similarity,  △ABC ~ △AMD

 

And we know   AM  =  c / 2   because  M  is the midpoint of  AB

 

So the scale factor from  △ABC  to  △AMD  is  1/2   And so...

 

AM  =  c / 2

AD  =  b / 2

DM  =  a / 2

 

Then by SAS congruence we can determine that  △ADM  ≅  △CDM   and so...

 

CM  =  AM

CM  =  c / 2

CM  =  (1/2)(AB)

hectictar Jul 7, 2019

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