Question

Using vectors prove that a quadrilateral is a rectangle if and
only if its diagonals are congruent and bisect each other.
​

Answer 1

Answer:

Using vectors the quadrilateral is a rectangle if and only if its diagonals are congruent and bisect each other.

Explanation:

Let ABCD be the quadrilateral

its diagonals are congruent and bisect each other.

|AB| = |BC| = |CD| = |DA|

Now

|CD| = |DC|

|DA| = |AD|

We know that,

AC = AB+AD

DB = AB-AD

To prove, AC⊥DB

⇒AC · DB = 0

So,

AC . DB = (  AB + AD) . (AB - AD)

We know that,

(a + b)(a - b) = |a|² - |b|²

⇒AC.DB = |AC|² - |AD|²

Since, ABCD is a square

So,

⇒|AB| = |AD|

⇒ AC.DB = 0

⇒AC ⊥ DB

Hence, quadrilateral is a rectangle if and only if its diagonals are congruent and bisect each other.

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