MHB Show Diagonals of Kite are Perpendicular

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The discussion focuses on proving that the diagonals of a kite are perpendicular. A user suggests labeling the intersection of diagonals AC and BD as point E, setting it as the origin, and placing points A, C, B, and D on the coordinate axes. Through vector analysis, it is shown that the lengths of segments AB and CB are equal, as are segments AD and CD, confirming the kite's properties. The dot product of the diagonals AC and BD is calculated to be zero, establishing their perpendicularity. The approach emphasizes the geometric arrangement of the points to support the proof.
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"AB- BC"? Do mean "AB= BD"? Personally, I would NOT "let B be the origin". I would label the point where AC and BD intersect "E" and let that be the origin. Then A is (-a, 0) for some number, a, and C is (a, 0). B is (0, b) and D is (0,-d) (d is not necessarily equal to b). Then AB= <0, b>- <a, 0>= <a, b> and CB= <0, b>- <a, 0>= <-a, b> so that AB and CB have the same length, $\sqrt{a^2+ b^2}$. AD= <0, -d>- <-a, 0>= <a, -d> and CD= <0, -d>- <a, 0>= <a, -d> so that AD and CD also have the same length, $\sqrt{a^2+ d^2}$. Yes this is a "kite".

The diagonals are AC and BD. AC= <-a, 0>- <a, 0>= <-2a, 0>. BD= <0, b>- <0, d>= <0, b- d>. The dot product of AC and BD is <-2a, 0>.<0, b-d>= (-2a)(0)+ (0)(b-d)= 0. Therefore the diagonals are perpendicular..
 
brinlin said:
Taking $B$ as the origin, let $A,C,D$ be represented by vectors $\def\v{\mathbf} \v a,\v c,\v d$. You are told that $AB = BC$, which says that $\v{a.a} = \v{c.c}$. Also, $CD = DA$, so that $(\v d - \v c)\v.(\v d - \v c) = (\v d - \v a)\v.(\v d - \v a)$. Using those equations, you want to show that $AC = BD$, or in other words $(\v c - \v a).\v d = \v 0$.
Country Boy said:
I would label the point where AC and BD intersect "E" and let that be the origin. Then A is (-a, 0) for some number, a, and C is (a, 0). B is (0, b) and D is (0,-d) ...
By choosing $A$ and $C$ to be on the $x$-axis, and $B$ and $D$ to be on the $y$-axis, you are assuming that $AC$ is perpendicular to $BD$, which is what you are supposed to be proving.
 
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