Question 112225
Rectangle ABCD shares BC with equilateral triangle BCE. The length of CD is 4 cm, and the length of DA is 3 cm. What is the length of BE? Make a diagram if possible 

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Thanks I don't understand how to do this problem.

{{{drawing(200,100,-1,8,-1,4,
rectangle(0,0,4,3),triangle(4,0,4,3,6.598076211,1.5),
locate(-.3,-.3,A),locate(3.9,-.3,B), locate(6.7,1.8,E),
locate(4,4,C),locate(-.3,4,D) )}}}

We are told that DA is 3 cm. So we put 3 on DA:

{{{drawing(200,100,-1,8,-1,4,
rectangle(0,0,4,3),triangle(4,0,4,3,6.598076211,1.5),
locate(-.3,-.3,A),locate(3.9,-.3,B), locate(6.7,1.8,E),
locate(4,4,C),locate(-.3,4,D), locate(-.5,2,3) )}}}

Now we know that CB and DA have the same length because
they are opposite sides of a rectangle. And since DA has
length 3 cm. so does CB,  So we put a 3 on CB:

{{{drawing(200,100,-1,8,-1,4,
rectangle(0,0,4,3),triangle(4,0,4,3,6.598076211,1.5),
locate(-.3,-.3,A),locate(3.9,-.3,B), locate(6.7,1.8,E),
locate(4,4,C),locate(-.3,4,D), locate(-.5,2,3), locate(3.6,2,3) )}}}
 
We are told that triangle BCE is equilateral, so all three
sides of it have the same length.  And since CB is a side
of the equilateral triangle BCE, both CE and BE have the
same length as CB. Since CB has length 3 cm, so doe BE, so
we put 3 on BE and CE.

{{{drawing(200,100,-1,8,-1,4,
rectangle(0,0,4,3),triangle(4,0,4,3,6.598076211,1.5),
locate(-.3,-.3,A),locate(3.9,-.3,B), locate(6.7,1.8,E),
locate(4,4,C),locate(-.3,4,D), locate(-.5,2,3), locate(3.6,2,3),
locate(5.3,.8,3),locate(5.3,2.9,3) )}}}

We didn't even need the fact that the length of CD is 4.

The answer is "the length of BE is 3 cm."

Edwin</pre>