1. Revision Description Reflect and Review Teasers Answers Recall of basics of triangles, polygons etc. Review Following are few examples of polygons:
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1 1. Revision Recall of basics of triangles, polygons etc. The minimum number of line segments required to form a polygon is 3. 1) Name the polygon formed with 4 line segments of equal length. 1) Square polygon with least number of sides is a triangle. 2. Polygons escription Polygons polygon is a closed figure made up of three or more line segments. Poly means many and polygon means a many-sided figure. Reflect and Review Following are few examples of polygons: Teasers 1) Name the following polygons having a) 6 sides b) 11 sides c) 10 sides nswers 1) a) Hexagon b) Hendecagon c) ecagon Number of vertices in a polygon = Number of sides in the polygon Some polygons have special names that correspond to the number of sides in a The minimum 1
2 lassification of Polygons Polygons are classified according to the number of sides they have. number of sides that a polygon can have is 3. Number of sides in a polygon Name Figure 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 ctagon 2
3 9 Nonagon 10 ecagon 11 Hendecagon 12 odecagon 3. iagonals of a Polygons diagonal is a line segment joining one vertex of the polygon to another, i.e., not next to it. For a polygon of n sides, the number of diagonals = ( ) For pentagon E,,,, E and E are its diagonals. 1) How many diagonals does each of the following have? a) nonagon b) dodecagon 1) a) 27 b) 35 To find the number of diagonals in a heptagon Number of diagonals =, since number of diagonals in a polygon with n sides is ( ). 3
4 4. Types of Polygons onvex Polygon In a polygon, if the line joining any two points on it lies inside the polygon, then the polygon is a convex The diagonals of a convex polygon lie entirely inside the is a concave polygon as some part of the diagonals lie outside the 1) heck whether the polygon given below in a convex or concave 1) oncave polygon 2) Irregular polygon oncave Polygon In a polygon, if there is even one diagonal which does not lie inside it completely, then the polygon is a concave oncave polygons have some portions of their diagonals in their exterior. n equilateral triangle is a regular polygon since all of its sides are of equal length. regular quadrilateral is known as a square. rectangle is equiangular but not equilateral. Thus, it is not a regular 2) Is the figure given below a regular or irregular Regular Polygon polygon in which all the sides and all the angles are 4
5 equal is called a regular Irregular Polygons The polygons which are not regular, i.e., whose sides and angles are not equal are called irregular polygons. 5. Quadrilaterals quadrilateral is a four-sided figure. Quadri means four and lateral means sides In the above Quadrilateral a. The points,, and are 7 To calculate the angles of quadrilateral whose ratio is 2 : 3 : 5 : 8. Let the angles be 2x, 3x, 5x and 8x. Sum of angles in a quadrilateral = x + 3x + 5x + 8x = x = So, the angles are 2x = 2(20) = ) In a quadrilateral, and = 75 0, also and are in the 5 : 4. Find the remaining angles. 2) In a quadrilateral the difference between the largest and smallest angle is 70 0 and the rest two angles are 1) = 105 0, = 84 0 and = ) 45 0, 100 0, 100 0,
6 called the vertices. b. The line segments,, d and are called the sides. c. The four angles 1, 2, 3 and 4 are interior angles. d. The four angles 5, 6, 7 and 8 are exterior angles. e. The sides and ; and ; and ; and are known as adjacent angles whereas and ; and are known as opposite sides of the quadrilateral. ngle Sum Property of Quadrilateral The sum of angles in a quadrilateral is 360 0, i.e., in quadrilateral, = x = 3(20) = x = 5(20) = x = 8(20) = equal and each 10 0 more than the smallest angle. Find all the angles. 6. Interior and Exterior ngle Properties of a Quadrilateral Interior angle property of a polygon In a regular polygon of n sides, Sum of interior angles of a regular angles = (n 2) Each interior angle of a regular polygon = ( ) To calculate each interior angle of a hexagon: n = 6 Each interior angle of hexagon = ( ) 1) Find the sum of interior angles of dodecagon. 1) Exterior angle property of a polygon In a regular polygon of n sides Sum of exterior angles of a To calculate each exterior angle of a octagon: n = 8 Each exterior angle of 1) Find each exterior angle of a regular pentagon. 1)
7 regular polygon = Each exterior angle of a regular polygon = octagon = Types of Quadrilateral 7. Trapezium Trapezium quadrilateral with one pair of opposite sides parallel is called a trapezium. is a trapezium in which. Isosceles Trapezium In a trapezium, if the non-parallel sides are equal, then it is called an isosceles trapezium. In a trapezium if and = 65 0 and = 80 0 then to find the other two angles: Since in a trapezium, adjacent angles are supplementary, we have + = and + = = = = and = = = ) If is an isosceles trapezium with, = and = 3 and = 2. Find all the sides of the trapezium if its perimeter is 72 cm. 1) = 27 cm, = 9 cm and = = 18 cm is an isosceles trapezium in which and =. 8. Kite quadrilateral with two pairs of adjacent sides equal but opposite sides unequal is called a kite. The perimeter of a kite is 24 cm. If two unequal adjacent sides are in the ratio of 5 : 3 then to find the lengths of sides: 1) Find the length of sides of a kite of perimeter 66 1) 12 cm, 21 cm, 21 cm and 12 7
8 is a kite in which = and = Let the lengths of unequal sides be 5x and 3x Perimeter = 24 cm 5x + 3x + 5x + 3x = 24 16x = 24 cm and whose unequal adjacent sides are in the ratio of 4 : 7. cm Properties i. iagonals of a kite intersect at right angle, i.e., ii. Shorter diagonal is bisected by longer diagonal, i.e., = iii. Longer diagonal divides the kite into two congruent triangles, i.e., iv. = v. = and = 5x = 5(1.5) = 7.5 cm nd 3x = 3(1.5) = 4.5 cm So, the lengths of sides are 7.5 cm, 4.5 cm, 4.5 cm and 7.5 cm 9. Parallelogram quadrilateral with both pairs of opposite sides is called a parallelogram. is a parallelogram in which,. Properties i. pposite sides of a parallelogram are parallel, i.e., and. ii. pposite sides of a parallelogram are equal, i.e., = and = iii. pposite angles are equal i.e., = and = To find the measure of angles of a parallelogram in which the adjacent angles are in the ratio of 4 : 5: Let the adjacent angles be 4x and 5x. We know that the adjacent angles of a parallelogram are supplementary, i.e., 4x + 5x = x = So, we get 4x = 4(20) = 80 0 and 5x = 5(20) = The measure of all the angles 1) In a parallelogram, if one angle is 15 0 more than twice its adjacent angle, then find all the angles of the parallelogram. 2) If the adjacent sides of a parallelogram are in the ratio of 7 : 5 and the sum of all sides of a parallelogram is 48 cm, find the measure of sides of the 1) 55 0, 125 0, 55 0 and ) 14 cm, 10 cm, 14 cm and 10 cm 8
9 iv. The sum of the measure of all its angles is 360º i.e., = 360º v. djacent angles are supplementary i.e., + = 180 0, + = 180 0, + = and + = vi. Its diagonals bisect each other i.e., = and = is 80 0, 100 0, 80 0 and parallelogram. 10. Rhombus rhombus is a parallelogram whose adjacent sides are equal. R S P Properties i. ll the sides of a rhombus are equal, i.e., PQ = QR = RS = PS. ii. The angles of a rhombus are bisected by the diagonals. iii. The diagonals of a rhombus are perpendicular bisectors of one another. iv. The diagonals of a rhombus are not equal. Q In a rhombus, if the diagonals and bisect at and = 60 0, then to calculate the measure of: We know, in a rhombus, the diagonals bisect each other perpendicularly, i.e., = 90 0 So, in + + = = = = 30 0 We also know that the diagonal of the rhombus is also the angular bisector of the angle at the vertex it is formed. = 2 = = ) is a rhombus with = etermine. 1)
10 11. Rectangle rectangle is a parallelogram in which every angle is a right angle. Properties i. pposite sides of a rectangle are equal, i.e., = and = ii. ll angles of a rectangle are right angles, i.e., = = = = 90º iii. The diagonals of a rectangle are equal, i.e., = iv. The diagonals of a rectangle bisect each other, i.e., = and = To find the diagonal of a rectangle whose sides are 12 cm and 5 cm: Length of the diagonal = 1) Find the length of the diagonal of a rectangle with sides 9 cm and 12 cm. 2) The sides of a rectangle are in the ratio of 5: 2. Find its sides if the perimeter is 56 cm. 1) 15 cm 2) 20 cm and 8 cm 12. Square square is a special type of rectangle with all its sides and angles equal. It is also a regular To find the diagonal of a square whose side is 4 cm: iagonal of a square = 1) Find the length of the diagonal whose side is 7 cm. 1) Properties i. ll the sides of a square are equal i.e., = = = ii. ll the angles of a square are right angles i.e., = = = = 90º iii. The diagonals of a square are equal i.e., = iv. The diagonals of a square are perpendicular bisectors of one another i.e., =, =,. 10
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