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In the given figure, in $Δ A B C, D and E$ are the mid-points of the sides BC and AC respectively. Find out relation between DE and AB.

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DE = \frac{1}{2} AB

DE = AB

DE = 3 AB

DE = 2 AB

answer is A.

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Detailed Solution

Given vertices of the triangle are

A 4, − 2, B 2, − 2, C − 6, − 1

.
And D and E are the middle points of BC and AC.
We know that,
If the coordinates of two points are

x 1, y 1

and

x 2, y 2

then the coordinate of the middle point of the two points will be

x 1 + x 2 2, y 1 + y 2 2

.
Then, the coordinates of point D and E are

2 − 6 2, − 2 − 1 2 = − 2, − 32 and 4 − 6 2, − 2 − 1 2 = − 1, − 32

respectively.
Now,
We know that, the distance between any two points in a Cartesian plane

M a, b and N c, d

is given by

M N = (c − a) 2 + (d − b) 2

.
Then,

D E = − 1 − (− 2) 2 + − 32 − − 32 2 ⇒ D E = 1 + 0 ⇒ D E = 1 ... (1)

And

A B = (2 − 4) 2 + (− 2 − (− 2)) 2 ⇒ A B = 4 + 0 ⇒ A B = 4 ⇒ A B = 2 ... (2)

From (1) and (2) we get,

D E = 12 A B

.
Therefore, the relation between DE and AB is

D E = 12 A B

.
Hence, option (1) is correct.

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