CBSE Class 8 Mathematics Practical Geometry Notes Set 12

Constructing Quadrilaterals

A quadrilateral has 4 sides, 4 angles and 2 diagonals, but only five independent measurements are needed to fix its shape and size. A rough sketch with the given measurements marked always comes first.

Types of Data

Four sides and one diagonal: the diagonal splits the figure into two triangles. Draw the triangle with three known sides (SSS), then locate the fourth vertex as the intersection of two arcs drawn from the ends of the diagonal.
Two diagonals and three sides: draw the triangle whose three sides are known, then find the fourth vertex using one side and one diagonal as arc radii.
Two adjacent sides and three angles: the fourth angle follows from the angle sum 360 degrees, so there are four angles; draw the base, then the angles at its ends, and mark the sides.
Three sides and the two angles between them: draw the middle side, make the two angles at its ends, and mark the other sides along the rays.
Special quadrilaterals: their properties supply extra data, e.g. a square has equal sides and right angles (draw ABC by SAS), a rhombus has diagonals that are perpendicular bisectors of each other, a parallelogram has diagonals that bisect each other and equal opposite sides, a rectangle has all angles 90 degrees.

Checks Before Constructing

Every triangle used must satisfy the triangle inequality: the sum of any two sides is greater than the third. Angles of a quadrilateral add up to 360 degrees. In a parallelogram adjacent angles are supplementary.

Worked example: in quadrilateral ABCD, A = 60, C = 105 and D = 90 degrees, so B = 360 - 255 = 105 degrees. In a rhombus with diagonals 7 cm and 5 cm the half-diagonals 3.5 cm and 2.5 cm meet at right angles, so each side is sqrt(18.5), about 4.30 cm.