+0  
 
0
171
1
avatar

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
 #1
avatar+88898 
+1

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

28 Online Users

avatar
avatar
avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.