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ago in Geometry and Algebra by (31.0k points)

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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1 Answer

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ago by (31.1k points)

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.

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