Question

four small villages are situated at positions (-1, 6), (1, 2), (5, 4), and (3, 8). the block development officer plans to set up a community center at the position (2, 5). what has prompted the block development officer to set up the community center at (2, 5)?

Answer 1

Answer:

Point (2, 5) lies on the intersection of the diagonals of the quadrilateral formed by the four villages. Thus, it is equidistant from the villages which has prompted the block development officer to set up the community centre at (2, 5).

Step-by-step explanation:

We can verify this by using the distant formula:

Let A (-1, 6), B (1, 2), C (5, 4), and D (3, 8).

$AB = \sqrt{{ (- 1 - 1)}^{2} + {(6 - 2)}^{2} } = \sqrt{4 + 16} = \sqrt{20} = 2 \sqrt{5}$

$BC = \sqrt{{ (1 - 5)}^{2} + {(2 - 4)}^{2} } = \sqrt{16 + 4} = \sqrt{20} = 2 \sqrt{5}$

$CD = \sqrt{{ (5 - 3)}^{2} + {(4 - 8)}^{2} } = \sqrt{4 + 16} = \sqrt{20} = 2 \sqrt{5}$

$AD = \sqrt{{ ( - 1 - 3)}^{2} + {(6 - 8)}^{2} } = \sqrt{16 + 4} = \sqrt{20} = 2 \sqrt{5}$

Now, diagonals

$AC = \sqrt{{ ( - 1 - 5)}^{2} + {(6 - 4)}^{2} } = \sqrt{36 + 4} = \sqrt{40} = 2 \sqrt{10} \\ BD = \sqrt{{ ( 1 - 3)}^{2} + {(2 - 8)}^{2} } = \sqrt{4 + 36} = \sqrt{40} = 2 \sqrt{10}$

We see that it is a square.

Now we can find the midpoint of these diagonals.

The midpoint of diagonal AC

$(x,y) = ( \frac{ - 1 + 5}{2} ,  \frac{6 + 4}{2} ) = ( \frac{4}{2} ,  \frac{10}{2} ) = (2, 5)$

Checking it we find that it's the point (2, 5) by midpoint formula.

Thus, it's a suitable point for the purpose.