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16 Sep 2014, 00:53
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Square \(ABCD\) is the base of the cube, and square \(EFGH\) is the cube's top face, with point \(E\) above point \(A\), point \(F\) above point \(B\), and so on. What is the distance between the midpoint of edge \(AB\) and the midpoint of edge \(EH\) if the area of square \(ABCD\) is 2?
A. \(\frac{1}{\sqrt{2} }\)
B. 1
C. \(\sqrt{2}\)
D. \(\sqrt{3}\)
E. \(2\sqrt{3}\)
A. \(\frac{1}{\sqrt{2} }\)
B. 1
C. \(\sqrt{2}\)
D. \(\sqrt{3}\)
E. \(2\sqrt{3}\)
16 Sep 2014, 00:53
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Official Solution:
Square \(ABCD\) is the base of the cube, and square \(EFGH\) is the cube's top face, with point \(E\) above point \(A\), point \(F\) above point \(B\), and so on. What is the distance between the midpoint of edge \(AB\) and the midpoint of edge \(EH\) if the area of square \(ABCD\) is 2?
A. \(\frac{1}{\sqrt{2} }\)
B. 1
C. \(\sqrt{2}\)
D. \(\sqrt{3}\)
E. \(2\sqrt{3}\)
Look at the diagram below:
Notice that \(Z\) is the midpoint of \(AD\). We need to find the length of the line segment \(XY\).
Now, since the area of \(ABCD\) is 2, the length of each edge of the cube is \(\sqrt{2}\)..
\(XZ=\sqrt{AX^2+AZ^2}=\sqrt{(\frac{\sqrt{2} }{2})^2+(\frac{\sqrt{2} }{2})^2}=1\);
\(XY=\sqrt{XZ^2+YZ^2}=\sqrt{1^2+(\sqrt{2})^2}=\sqrt{3}\).
Answer: D
Square \(ABCD\) is the base of the cube, and square \(EFGH\) is the cube's top face, with point \(E\) above point \(A\), point \(F\) above point \(B\), and so on. What is the distance between the midpoint of edge \(AB\) and the midpoint of edge \(EH\) if the area of square \(ABCD\) is 2?
A. \(\frac{1}{\sqrt{2} }\)
B. 1
C. \(\sqrt{2}\)
D. \(\sqrt{3}\)
E. \(2\sqrt{3}\)
Look at the diagram below:
Notice that \(Z\) is the midpoint of \(AD\). We need to find the length of the line segment \(XY\).
Now, since the area of \(ABCD\) is 2, the length of each edge of the cube is \(\sqrt{2}\)..
\(XZ=\sqrt{AX^2+AZ^2}=\sqrt{(\frac{\sqrt{2} }{2})^2+(\frac{\sqrt{2} }{2})^2}=1\);
\(XY=\sqrt{XZ^2+YZ^2}=\sqrt{1^2+(\sqrt{2})^2}=\sqrt{3}\).
Answer: D
01 Sep 2016, 07:32
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I agree with explanation. easier way to answer this Q. can be
We know that distance is more than an edge which is 2^(1/2) but less than longest diagonal which is (3*2)^(1/2) so it has to be 3^(1/2)
We know that distance is more than an edge which is 2^(1/2) but less than longest diagonal which is (3*2)^(1/2) so it has to be 3^(1/2)
