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Initially I was of the opinion that if two diagonals in a parallelogram intersect then the angle formed at the point of intersection is 90 degrees (I came up with this conclusion by inserting values) , if this applied to a parallelogram then I assumed it also applied to a square or a rectangle.In both the figures below i assumed angles A , B , C will always be 90 degrees at least that's what I thought.Am i wrong ?

enter image description here

Now I just came across a question which is conflicting with this concept the question is:

ABCD is rectangle the diagonals AC and BD intersect at E (reflected in red figure) .Which of the statement is not necessarily true. The answer to this was : AE is perpendicular to BD. Could anyone let me know why is this not true ? Did i have the wrong idea to start with ?

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1 Answer 1

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Draw (with a ruler) a long skinny rectangle, say base $6$, height $1$. Now draw the diagonals. You will see that the two diagonals definitely do not meet at right angles, not even close!

It turns out that a parallelogram has its diagonals meeting at right angles if and only if the parallelogram is a rhombus (all sides equal). Note that a square is a special case of a rhombus.

Proof: It is fairly easy to prove that the diagonals of a parallelogram (and therefore of the special parallelogram called a rectangle) bisect each other.

Let the two diagonals of a parallelogram have length $2p$ and $2q$ respectively. If the angle at which they meet is $90^\circ$, then by the Pythagorean Theorem each side of the rectangle has length $\sqrt{p^2+q^2}$. So in particular all the sides of the parallelogram are equal, that is, we have a rhombus.

Conversely, by a congruent triangles argument, if we have a rhombus, its diagonals meet at right angles.

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What about parallelograms ? Square ? –  MistyD Jun 19 '12 at 3:20
    
" It turns out that a parallelogram has its diagonals meeting at right angles if and only if the parallelogram is a rhombus (all sides equal)" - Thanks for clearing this up –  MistyD Jun 19 '12 at 3:26

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