Question

M and N are the mid-points of the diagonals AC and BD respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to ______.

पर्याय

• 2 MN

• 2 NM

• 4 MN

• 4 NM

Answer 1

M and N are the mid-points of the diagonals AC and BD respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to 4 MN.

Explanation:

Suppose that the position vectors of A, B, C, D, M and N are a, b, c, d, m and n respectively.

As, M and N are the mid-points of AC and BD.

So, m = `(a + c)/2` and n = `(b + d)/2`

Then, AB + AD + CB + CD

= (b – a) + (d – a) + (b – c) + (d – c)

= 2(b + d) – 2(a + c)

= 2 × 2n – 2 × 2m = 4(n – m) = 4MN

`\implies` 4 λ + 1 – 7 – 2 – λ = 10

`\implies` 3 λ = 18

`\implies` λ = 6