The diagonals of a convex quadrilateral are m and n. The segments connecting the midpoints of the opposite

| June 21, 2021 | education

The diagonals of a convex quadrilateral are m and n. The segments connecting the midpoints of the opposite sides are equal to each other. Prove that its area is 0.5mn.

The area of a convex quadrilateral is S = ½ * m * n * sinβ.

Find the angle β between the diagonals of the quadrangle.

Let us denote the convex quadrilateral ABCD, and the midpoints of the sides AB, BC, CD, AD by the letters K, L, M, N.

Moreover, according to the condition KM = NL.

Then, KLMN is a rectangle since its diagonals KM and NL are equal.

The sides KL and MN, NK and ML of the rectangle KLMN are the midlines of the triangles ABC and ADC, DAB and DCB, respectively, and therefore are parallel to the diagonals AB and BD.

Therefore, the diagonals of the quadrilateral are perpendicular to each other and the angle β = 90º.

Then, S = ½ * m * n * sin90 = 0.5 * m * n.



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