SECTION III. QUADRILATERAL FIGURES. DEFINITIONS. 1. A quadrilateral figure is one which is bounded by four straight lines. Ex. ABCD B 2. A parallelogram is a quadrilateral, of which the opposite sides are parallel and equal. Ex. ABCD B A NOTE.-Quadrilaterals fall into six classes, four of which are parallelograms-viz., the square, the rectangle, the rhombus, and the rhomboid. 3. A square is a parallelogram which has all its sides equal, and all its angles right angles. Ex. ABCD. B 4. A rectangle is a parallelogram which has only its opposite sides equal, but all its angles right angles. Ex. ABCD B A NOTE. This kind of parallelogram is also termed an oblong. 5. A rhombus is a parallelogram which has all its sides equal, but its angles are not right angles. Ex. ABCD NOTE.-In each case its opposite angles are equal to each other. 6. A rhomboid is a parallelogram which has only its opposite sides equal, but its angles are not right angles. B Ex. ABCD C NOTE 1.-As in the preceding figure, its opposite angles are equal to each other. NOTE 2.-The remaining classes of quadrilaterals are the trapezium and the trapezoid. 7. A trapezium is a quadrilateral which has none of its sides parallel. Ex. ABCD D 8. A trapezoid is a quadrilateral which has only two of its sides parallel. Ex. ABCD— A NOTE 1.-Some of its sides and angles may be equal. NOTE 2.-All quadrilateral figures are also called quadrangles, as they have also four angles. 9. A diagonal of a quadrilateral is a straight line which joins any two of its opposite angles. Ex. AB— 10. A diameter is a straight line drawn through its centre parallel to two of its sides. Ex. AB— A B Problem 34. To construct a square on a given base AB— 1. At B in the given line AB erect a perpendicular BC equal to AB (Pr. 2). 2. From the point A, with radius AB, describe an arc above A. 3. From C, with the same radius, cut the arc in D. D A 4. Join AD and CD. Then ABCD is the square required. Problem 35. To construct a square, the diagonal AB being given. 1. Bisect AB by the perpendicular CD (Pr. 1). 2. Cut off EF, and EG, equal to EA or EB. 3. Join AF, FB, BG, GA. Then AFBG is the square required, having the given diagonal AB. Problem 36. To construct a rectangle, the lengths of two of the sides AB and BC being given. 1. From the point B in the given line AB, erect BC perpendicular to AB (Pr. 2), and equal to the given line BC. 2. From A, with radius BC, describe an arc above A. 3. From C, with radius AB, cut the arc in the point D. Problem 37. To construct a rectangle, one side AB and its diagonal AC being given. 1. Bisect the diagonal AC in E (Pr. 1), and from E as centre, with the radius EA or EC, describe a circle. 2. From A and C as centres, with the given line AB as radius, describe arcs B and D. 3. Join AB, BC, CD, and DA. Then ABCD is the rectangle required. |