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Find the lengths of the medians of the triangle whose vertices are $(5, 6), (3, 8)$ and $(- 1, 2)$ .

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17 units, 52 units, 17 units

17 units, 52 units, 11 units

17 units, 53 units, 17 units

19 units, 52 units, 17 units

answer is A.

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Detailed Solution

Given vertices of a triangle are

5, 6, 3, 8, − 1, 2

From figure,
The medians AD, BE, and CF are drawn from vertices A, B, and C respectively.
So points D, E, and F are the midpoint of BC, AC, and AB respectively.
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 co-ordinate of point D is

3 − 1 2, 8 + 2 2 = (1, 5)

.
The co-ordinate of point E is

5 − 1 2, 6 + 2 2 = (2, 4)

.
And,
The co-ordinate of point F is

5 + 3 2, 8 + 6 2 = (4, 7)

.
As the length of the median AD is the distance between point A and D, and we know that, the distance between any two points in a Cartesian plane

M a, b, N c, d

is given by

M N = (c − a) 2 + (d − b) 2

.
Then,

A D = (1 − 5) 2 + (5 − 6) 2 = 16 + 1 = 17

Similarly,

C F = (4 − (− 1)) 2 + (2 − 7) 2 = 25 + 25 = 50 = 52

And

B E = (2 − 3) 2 + (4 − 8) 2 = 1 + 16 = 17

Therefore, the length of the medians is

17 units, 52 units, 17 units

.
Hence, option (1) is correct.

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