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Right triangle  has one leg of length 6 cm, one leg of length 8 cm and a right angle at . A square has one side on the hypotenuse of triangle  and a vertex on each of the two legs of triangle . What is the length of one side of the square, in cm? Express your answer as a common fraction.
 

Guest Mar 11, 2018
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Let two vertexes of the square lie on  (a,0) and (0, b)

Where a < 8    and b < 6

 

Let side BC be the hypotenuse of the right triangle

And the equation of BC is:

y  =  -(6/8)(x - 8)

8y = -6x + 48

6x + 8y - 48 = 0

 

Now....using the formula for the distance between a point and a line we have two outcomes

 

Distance between (a,0)  and the line  =

 

abs [6(a) + 8(0) - 48 ]           abs[    6a  -  48]          abs[ 3a  - 24]

_________________      =    ____________      =   _________

     10                                        10                                    5

 

Distance between (0,b) and the line =

 

 abs[6(0) + 8(b -48) ]         abs[  8b - 48]              abs [ 4b - 24]

 ________________      =  ___________    =      ___________

     10                                       10                                5

 

And these distances are equal

 

Since a < 8  and b < 6 we can write

 

(24 - 3a) / 5  = (24 - 4b) / 5

24 - 3a  = 24 - 4b

-3a  = - 4b

3a = 4b  ⇒  b = (3/4) a

 

And the distance between the two points = the distance from one of the points to the hypotenuse 

 

Thus...

 

√[a^2 + b^2]  =  abs (3a - 24 )/ 5

 

Since  a < 8,  and the left side must be > 0...we can write...

 

√[a^2 + b^2]  =   (24 - 3a ) / 5

√ [a^2 + (9/16)a^2 ]  =  (24 - 3a) / 5

√ [ 25a^2] / 4  =  (24 - 3a)/5

5a/4  =  (24 - 3a) / 5

25a/4 = 24 - 3a

25a  = 96 - 12a

37a = 96

a = 96/37     

b =  (3/4 a)   = (3/4)(96/37) =   72/37

 

So.....the length of the side of the square  =

 

√[(96/37)^2 + (72/37)  ^2]  =

 

√[96^2 + 72^2] / 37   =  √14400 /  37    =   120 / 37 cm 

 

Here's a pic.... D  = "a"  = (96/37, 0)   and  E = "b"  = (0, 72/37)

 

 

 

cool cool cool

CPhill  Mar 12, 2018

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