The trapezium, its midline, a circumscribed quadrilateral and the area of a trapezium
Learn the properties of an isosceles trapezium, the midline, the criterion for a quadrilateral with an inscribed circle and the area of a trapezium.
- Ukraine: grade 8
Guide
In the Ukrainian programme this is Grade 8 geometry, §1: item 8 "The trapezium", item 10 "Circumscribed and inscribed circles of a quadrilateral" and item 23 "The area of a trapezium".
A trapezium has exactly one pair of parallel sides (as in the Ukrainian books). The parallel sides are the bases, the others are the legs. In an isosceles trapezium the legs are equal, the angles at a base are equal and the diagonals are equal. The midline joins the midpoints of the legs, is parallel to the bases and equals half their sum.
A circle can be inscribed in a convex quadrilateral if and only if the sums of opposite sides are equal: AB + CD = BC + AD.
The area of a trapezium is S = a + b2·h, where h is the height, not the leg.