CBSE Class 10 Mathematics Quadrilaterals Notes Set 14

Quadrilaterals

A quadrilateral has four sides, four angles and two diagonals, and the sum of its angles is 360°. A parallelogram has both pairs of opposite sides parallel; a rectangle, rhombus and square are special parallelograms.
Parallelogram: opposite sides equal, opposite angles equal, adjacent angles supplementary, diagonals bisect each other. A quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel, or if the diagonals bisect each other.
Rectangle: parallelogram with all angles 90°; its diagonals are equal. Rhombus: parallelogram with all sides equal; its diagonals are perpendicular and bisect the angles. Square: both a rectangle and a rhombus.
Trapezium: one pair of parallel sides. In an isosceles trapezium the non-parallel sides are equal.

Mid-point Theorem

The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. Conversely, a line through the mid-point of one side parallel to another side bisects the third side.
Joining the mid-points of the sides of any quadrilateral in order gives a parallelogram. The segment joining the mid-points of the non-parallel sides of a trapezium is parallel to the parallel sides and equals half their sum; the segment joining the mid-points of its diagonals is half their difference.

Area of Parallelogram and Triangle

Area of a parallelogram = base × corresponding height. Parallelograms on the same base and between the same parallels have equal area. Area of a triangle = ½ × base × height, and a triangle on the same base and between the same parallels as a parallelogram has half its area.
Area of a rhombus = ½ × (product of its diagonals). Area of a trapezium = ½ × (sum of parallel sides) × height.

Polygons

1. Sum of the interior angles of a convex polygon of n sides = (n − 2) × 180°.
2. Sum of the exterior angles (one at each vertex) = 360°.
3. In a regular polygon each interior angle is (n − 2) × 180°/n and each exterior angle is 360°/n.
4. Number of diagonals = n(n − 3)/2.

Worked Examples

Illustration 1. The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find them.
Answer:
10k = 360°, so k = 36°.
Angles: 36°, 72°, 108°, 144°.

Illustration 2. Find the number of sides of a regular polygon each of whose exterior angles is 24°.
Answer:
n = 360°/24° = 15.

Illustration 3. The diagonals of a rhombus are 16 cm and 12 cm. Find its side and area.
Answer:
Half-diagonals are 8 cm and 6 cm, so the side = √(64 + 36) = 10 cm.
Area = ½ × 16 × 12 = 96 cm².