In a convex quadrilateral ABCD, the length of the segment connecting the midpoints of sides AB and CD is 1.

| May 23, 2021 | education

In a convex quadrilateral ABCD, the length of the segment connecting the midpoints of sides AB and CD is 1. Lines BC and AD are perpendicular. Find the length of the line segment connecting the midpoints of the diagonals AC and BD.

Decision:

1) Let O be the midpoint of side AB, and point L is the midpoint of the side CD given to us in the condition of the convex quadrilateral ABCD.

2) Let point I be the midpoint of the diagonal AC of the quadrangle ABCD, and point V the midpoint of the diagonal BD. Then we get that OI is the midline of triangle ABC (by definition of the midline), and LV is the midline of triangle DBC (by definition).

3) We get: OI = BC = LV and OI || BC || LV.

4) Based on point 3, we get that the quadrilateral OILV is a parallelogram. OI and OV are parallel to BC and AD, so OI is perpendicular to OV. Therefore, the quadrilateral OILV is a rectangle (by property). The diagonals of the rectangle are equal (by property), so IV = OL = 1.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.