The coordinates for a rhombus are given as (2a, 0) (0, 2b), (-2a, 0), and (0.-2b). How do you write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry?

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The coordinates for a rhombus are given as (2a, 0) (0, 2b), (-2a, 0), and (0.-2b). How do you write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry?

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Answer 1 · 2017-01-12T17:11:52

Please see below.

Explanation:

Let the points of rhombus be $A (2 a, 0), B (0, 2 b), C (- 2 a, 0)$ $A(2a, 0), B(0, 2b), C(-2a, 0)$ and $D (0 . - 2 b)$ $D(0.-2b)$ .

Let midpoints of $A B$ $AB$ be $P$ $P$ and its coordinates are $(\frac{2 a + 0}{2}, \frac{0 + 2 b}{2})$ $((2a+0)/2,(0+2b)/2)$ i.e. $(a, b)$ $(a,b)$ . Similarly midpoint of $B C$ $BC$ is $Q (- a, b)$ $Q(-a,b)$ ; midpoint of $C D$ $CD$ is $R (- a, - b)$ $R(-a,-b)$ and midpoint of $D A$ $DA$ is $S (a, - b)$ $S(a,-b)$ .

It is apparent that while $P$ $P$ lies in Q1 (first quadrant), $Q$ $Q$ lies in Q2, $R$ $R$ lies in Q3 and $S$ $S$ lies in Q4.

Further, $P$ $P$ and $Q$ $Q$ are reflection of each other in $y$ $y$ -axis, $Q$ $Q$ and $R$ $R$ are reflection of each other in $x$ $x$ -axis, $R$ $R$ and $S$ $S$ are reflection of each other in $y$ $y$ -axis and $S$ $S$ and $P$ $P$ are reflection of each other in $x$ $x$ -axis.

Hence $P Q R S$ $PQRS$ or midpoints of the sides of a rhombus $A B C D$ $ABCD$ form a rectangle.
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The coordinates for a rhombus are given as (2a, 0) (0, 2b), (-2a, 0), and (0.-2b). How do you write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry?

1 Answer

Explanation:

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