Question

Using vectors prove that a quadrilateral is a rectangle if and only if its diagonals are congruent and bisect each other.
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Answer 1

Answer:

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

​Step-by-step explanation:

Let ABCD be a quadrilateral

This quadrilateral is square.

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

Now

|CD| = |DC|

|DA| = |AD|

the diagnoal are AC and DB

it is square.

then the diagonals bisect each other

We have to prove that AC ⊥ DB

From square law of addition

We know that,

AC = AB+AD and

DB = AB-AD

To prove that 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 proved.

Answer 2

ANSWER:

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

Hope It Will Help You!