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If the rectangular faces of a brick have their diagonals in the ratio 3 : 2 \sqrt{3} : \sqrt{15}, what is the ratio of the length of the shortest edge of the brick to that of its longest edge?

 Jul 12, 2020
 #1
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Let a, b, and c be length of the three sides of the brick.

From the ratios given, we can write these equations.
(1) (a^2+b^2)=(3)^2=9
(2) (b^2+c^2)=(2√3)^2=12
(3) (a^2+c^2)=(√15)^2=15

(1)+(2)+(3)
2*(a^2+b^2+c^2)=36
(a^2+b^2+c^2)=18
(9+c^2)=18
c^2=9

(2)
(b^2+c^2)=12
(b^2+9)=12
b^2=3

(1)
(a^2+b^2)=9
(a^2+3)=9
a^2=6

 

(I put too much effort into this... why didn't I just use sqrt(#) whatever)

 

The longest and shortest are c, b.

 

b^2:c^2=3:9=1:3

 

So B:C=1:√3

 

Yay!

 Jul 12, 2020

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