# The coordinates of the vertices of a quadrilateral, taken in order, are (2, 1), (5, 3), (8, 7), (4, 9).

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The coordinates of the vertices of a quadrilateral, taken in order, are (2, 1), (5, 3), (8, 7), (4, 9).

i. Find the coordinates of the midpoints of all four sides.

ii. Prove that the quadrilateral got by joining these midpoints is a parallelogram.

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i. Mid point of AB

$(\frac{2+5}{2},\frac{1+3}{2})$

$(\frac{7}{2},\frac{4}{2})=(\frac{7}{2},2)$

= P(3,5,2)

= S(3,5)

ii. Length of PQ

Length of QR

Length of RS = $\sqrt{(6-3)^2+(8-5)^2}$

$=\sqrt{3^2+3^3}=\sqrt{18}$

Length of PS = $\sqrt{(3.5-3)^2+(2-5)^2}$

$=\sqrt{5^2+3^2}=\sqrt{9.25}$

PQ = RS

RQ = PS

Since the opposite sides of the quadrilateral PQRS are equal it is a parallelogram.