+0  
 
0
256
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+78618 
+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
Sort: 

2+0 Answers

 #1
avatar+78618 
+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+91001 
+5

Nice work chris,  Your diagram looks good :)

Melody  Aug 24, 2015

20 Online Users

avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details