+0  
 
0
441
2
avatar+238 

abcd is a square .from the diagonal bd, a length bx is cut off equal toba.from x, a straight line xy is drawn perpendicular to bd to meet ad at y.then ab+ay=

matsunnymat  Aug 23, 2015

Best Answer 

 #1
avatar+86859 
+15

Call the side of the square S.....

 

Using the Law of Cosines, we have

 

AX^2 = 2S^2 -2S^2cos(45)   =  2S^2  - 2S^2(1/√2)  =  S^2 [ 2 - √2]

 

So........AX =  S*√[ 2 - √2]

 

And using some basic geometry <YXD = 90  and <XDA = 45......so <XYD = 45....so <AYX = [180- <XYD]= 135

 

And since AB = BX and <ABD = 45, then <AXB  = (180 - 45]/2 = 67.5

 

Then <AXY = [180 - 90 - 67.5] = 22.5

 

And using the  Law of Sines, again, we have

 

AY/sin(22.5) = AX/sin(135)

 

AY = sin(22.5)/sin(135)* S*√[ 2 - √2] = [√[1-1/√2] / √2] * √2* √[2 - √2]S  = [√[1-1/√2]*√[2 - √2]*S  = [ √[√2 -1] * √[2 - √2] / √2]*S =[√ [2√2 - √2 - 2 +√2] / √2]*S  = [ √[2√2 -2]/ √2]*S= [√[(2)(√2 -1 )] / √2] *S   = [√2 - 1]*S

 

So.... AB + AY = S + [√2 - 1]S  = S [ 1 + [√2  - 1] ] S  = √2S

 

Her's an (aproximate) picture......

 

 

 

CPhill  Aug 23, 2015
 #1
avatar+86859 
+15
Best Answer

Call the side of the square S.....

 

Using the Law of Cosines, we have

 

AX^2 = 2S^2 -2S^2cos(45)   =  2S^2  - 2S^2(1/√2)  =  S^2 [ 2 - √2]

 

So........AX =  S*√[ 2 - √2]

 

And using some basic geometry <YXD = 90  and <XDA = 45......so <XYD = 45....so <AYX = [180- <XYD]= 135

 

And since AB = BX and <ABD = 45, then <AXB  = (180 - 45]/2 = 67.5

 

Then <AXY = [180 - 90 - 67.5] = 22.5

 

And using the  Law of Sines, again, we have

 

AY/sin(22.5) = AX/sin(135)

 

AY = sin(22.5)/sin(135)* S*√[ 2 - √2] = [√[1-1/√2] / √2] * √2* √[2 - √2]S  = [√[1-1/√2]*√[2 - √2]*S  = [ √[√2 -1] * √[2 - √2] / √2]*S =[√ [2√2 - √2 - 2 +√2] / √2]*S  = [ √[2√2 -2]/ √2]*S= [√[(2)(√2 -1 )] / √2] *S   = [√2 - 1]*S

 

So.... AB + AY = S + [√2 - 1]S  = S [ 1 + [√2  - 1] ] S  = √2S

 

Her's an (aproximate) picture......

 

 

 

CPhill  Aug 23, 2015
 #2
avatar+92623 
+5

Nice work chris,  Your diagram looks good :)

Melody  Aug 24, 2015

14 Online Users

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.