Ganit Learning Guides. Basic Geometry-2. Polygons, Triangles, Quadrilaterals. Author: Raghu M.D.

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1 Ganit Learning Guides asic Geometry-2 Poygons, Trianges, Quadriateras uthor: Raghu M..

2 ontents GEOMETRY... 2 POLYGONS... 2 Trianges... 2 Quadriateras asic-geometry2 1 of ,

3 GEOMETRY POLYGONS Poygon is a cosed figure with intersecting ines forming vertices. The ine segments form sides of poygons. They do not cross over or extend beyond the vertices. The simpest of Poygons is a triange with ony three sides. Poygons can have any number of sides. Poygon with infinite number of sides wi have a shape which is cose to a irce. If the sides are of the same size, then the figure is caed a reguar poygon. Triange Hexagon Irreguar Poygon Trianges Triange has three sides, three anges and three vertices. It is named by the three vertices, for exampe as. c b a has three vertices, and, three sides, and, and three anges, and. The sides of the triange can aso be abbreviated as a, b and c. Here a is the side opposite to vertex, b is the side opposite to vertex and c is the side opposite to vertex. The anges can aso be abbreviated as, and. asic-geometry2 2 of ,

4 Types of Trianges Trianges are cassified according to their shapes. Scaene trianges Fig.(a) Fig.(b) These are trianges with different engths of sides and different anges. the three anges can be acute as shown in Figure (a) or one of them can be obtuse as shown in Figure (b). Right nge triange One of the anges of a Right nge triange is 90º. Isoscees triange Two sides and two anges of an Isoscees triange are equa to one another. Equiatera triange the three sides and three anges of an Equiatera triange are equa to one another. Properties of Trianges c h a b asic-geometry2 3 of ,

5 Perimeter of a triange The Perimeter P of a triange of sides a, b and c is a + b + c, where a, b and c are engths of sides opposite to the vertices, and. P = a + b + c rea of a triange onsider a triange as shown above. Side b is taken as the base of the triange, h is the perpendicuar distance of the vertex to the base, which is aso considered as the height of the triange. rea Or = ½ b h = ½ bh Sum of anges The three interior anges of a triange add up to 180º. Proof: onstruction: raw a. raw a ine E parae to passing through the vertex. E = E = (aternate anges) (aternate anges) Hence, + E = + dding to both sides, + E + = + + ut, + + E = 180º (anges on a ine) asic-geometry2 4 of ,

6 oncusion: + + = 180º The interior anges of a triange add up to 180º. Exampe 1: Show that each ange in an equiatera triange is equa to 60º. onstruction: raw a triange with equa sides using a ruer and mark the interior anges as aº. aº aº aº In an equiatera triange a the three anges are equa. Hence, the sum of the anges, aº + aº + aº = 180º (anges in a triange add up to 180º) 3aº = 180º Or. aº = 180/3 = 60º nswer: Each ange in an equiatera triange is equa to 60º. Exampe 2: Triange is an isoscees triange. The perimeter of is 13cms amd the shortest side is 3cms. Find the engths of the remaining two sides. onstruction: raw the figure of an isoscees and mark the shortest side as 3cms. 3 asic-geometry2 5 of ,

7 Perimeter P = + + Or + + = 13 (given P = 13) + + = 13 ( =, is isoscees) = 13 (given = 3) 2 = 10 or = 10/2 = 5 = = 5 nswer: = = 5cms. Exampe 3: PQR is a right ange triange, where PQ = 3cms, QR = 4cms and RP = 5cms. Find the area of PQR. onstruction: raw a right ange triange with sides PQ, QR and RP, where PQR = 90º. P 3 5 Q 4 In a right ange triange, the two mutuay perpendicuar sides can be considered as the base and height of the triange. In PQR, PQ QR ( PQR = 90º) R rea = ½ base height = ½ QR PQ = ½ 4 3 = 6 nswer: rea of PQR = 6 cms 2 asic-geometry2 6 of ,

8 Exampe 4: ine of is extended to point and = = 60º. Find the vaue of. onstruction: Sketch a and extend the side to a point. Mark. In, + + = 180º (anges in a ) + = 180º (anges on a ine) Hence, ( + + ) ( + ) = 180º - 180º = 0 60º + 60º = 0 = 120º nswer: = 120º Exampe 5: In the figure beow, the ine segment E is parae to of. If = 70º and = 50º, find the vaue of, E and E. E onstruction: Sketch a and draw a ine segment E parae to. + + = 180º (anges in a triange) asic-geometry2 7 of ,

9 70º + 50º + = 180º (given = 70º, = 50º) 120º + = 180º = 180º 120º = 60º = 60º ut, E = (given E ) E = = 70º (given = 70º) E = 70º Simiary E = (given E ) E = = 60º (by cacuation = 60º) E = 60º nswer: = 60º, E = 70º and E = 60º. EXERISE Trianges asicgetri 1. State whether True of Fase: a) In a, = + b) Three anges of a triange add up to 180º. c) In a PQR, P is acute and Q and R are obtuse. d) rea of a triange is given by the formua: = ½ bh. 2. n equiatera triange has a perimeter of 21cms. Find the ength of each side. 3. Find the area of a scaene triange with base 5cms and height of 4cms. 4. is shown in figure beow. Find the vaue of xº and yº, if the = 90º. xº 2xº yº asic-geometry2 8 of ,

10 5. Find the perimeter of a PQR, if the ength of side PR = 8cms and Q = R = 60º. 6. Find the size of anges of shown beow, given that = xº+40º, = xº and = xº 10º. 7. Show that the exterior ange of a triange is equa to the sum the two interior opposite anges. 8. In the figure shown beow, Line E is parae to side. Find the size of and E. E 60º 40º 9. Using a ruer and protractor, draw the figure of such that = 40º, = 60º and = 6cms. Measure using the protractor. 10. Find the sizes of anges marked xº and yº. xº xº 60º 80º yº NSWERS 1. a) F b) T c) F d) T 2. 7cms asic-geometry2 9 of ,

11 3. 20 sq.cms 4. x = 30º, y = 120º 5. Perimeter = 24cms 6. = 90º, = 50º and = 40º 7. Hint: nges in a triange and anges on a ine add up to 180º. 8. = 100º and E = 40º º ± 1º 10. x = y = 40º Quadriateras Quadriatera is a four-sided poygon, with four sides, four anges and four vertices. In this figure,, and are the vertices.,, and are the sides. Lines joining, and, are diagonas of the quadriatera.,, and are interior anges. In any quadriatera, the sum of interior anges is equa to 360º. Proof: onsider two trianges and in the quadriatera shown above. + + = 180º + + = 180º (anges in a triange) (anges in a triange) asic-geometry2 10 of ,

12 = 180º + 180º + + ( + ) + ( + ) = 360º = 360º oncusion: The interior anges of a quadriatera add up to 360º. Specia Quadriateras Quadriateras can be cassified according to their shapes. Some of them are specia because of their unique properties such as parae sides, equa anges, etc. Using these properties, the engths of sides, size of anges, perimeter and area can be cacuated. Square Properties Equa sides of ength. Equa interior anges of size 90º. Perimeter = 4 rea = 2 iagonas bisect the interior anges and are of equa ength. Rhombus Properties Equa sides. Pairs of equa opposite anges. Perimeter = 4 rea = Product of the engths of diagonas iagonas intersect at their midpoints. b Rectange Properties Opposite sides are equa and parae. Equa interior anges of size 90º. Perimeter = 2 + 2b rea = b Paraeogram Square, rectange and rhombus are a types of paraeograms. Properties Opposite sides are equa and parae. Pairs of equa opposite anges. Perimeter = Sum of the engths of a sides rea = Product of the height (distance between two sides) and the ength of one of the remaining side. asic-geometry2 11 of ,

13 Trapezium trapezium has a pair of parae sides joined by unequa parae sides. Properties rea = Product of the height and the average ength of the two parae sides. Kite Properties Two pairs of adjacent equa sides. One pair of equa opposite anges. rea = Product of the engths of the diagonas Exampe 1: side of a square measures 5cms. Find its perimeter and area. onstruction: raw a square of side 5cms. =5 Perimeter P = 4 = 4 5 (give = 5cms) P = 20cms rea = 2 = 5 5 = 25 = 25 sq.cms nswer: Perimeter = 20cms, rea = 25cms. Exampe 2: The sum of engths of two adjacent sides of a rectange is 8cms. Its ength is 5cms. Find its perimeter and area. onstruction: raw a rectange and mark its ength as and breadth as b. + b = b = 8 b = 8 5 =5 b asic-geometry2 12 of ,

14 b = 3 Perimeter P = 2 ( + b) P = 2 8 = 16 rea = b = 5 3 = 15 nswer: Perimeter = 16cms, rea = 15sq.cms Exampe 3: The perimeter of a rhombus is 20cms and one of its anges is 120º. Find the vaues of remaining anges and ength of the sides. 120º onstruction: raw a rhombus. Mark = 120º. Perimeter P = 4 20 = 4 = 20/4 = 5 = 120º = = 120º (pair of opposite anges) ut = 360º (anges in a quadriatera) 120º º + = 360º + = 360º - 120º - 120º = 120º ut = (pair of opposite anges) + = 120º 2 = 120º = 120/2 = 60º = nswer: Length of each side = 5cms, = 60º, = 120º, = 60º. asic-geometry2 13 of ,

15 Exampe 4: Foowing is a diagram of a kite. Given = 110º and = 80º, find anges and. 80º 110º = = 110º = 360º 110º + 80º + 110º + = 360º 300º + = 360º (pair of equa opposite anges) (anges in a quadriatera) = 360º - 300º - 60º nswer: = 110º, = 60º. EXERISE Quadriateras asicgequad 1. State whether True or Fase: a) In a quadriatera, diagonas intersect and cross over. b) In a quadriatera, sides intersect and cross over. c) Sum of interior anges of a quadriatera add up to 360º. d) In a kite, opposite sides are parae. 2. Two pairs of opposite sides are parae. Name the quadriatera. 3. is a square. Show that the diagona divides the into two equa parts. 4. Find the size of xº in the foowing figure. = x + 20 = x = x 20 = x + 40 xº asic-geometry2 14 of ,

16 5. trapezium has two sides and equa to 5cms. The engths of two parae sides and are 8cms and 4cms respectivey. Find its perimeter. 6. PQRS is a paraeogram. Interior P = 60º and P = 120º. Find the anges R and S. 7. The foowing figure shows a kite. Its diagonas and intersect at point E. Given E = 2cms, = 3cms and = 2cms, find the area of. E 8. PQR is an isoscees triange. Line ST is parae to base QR. ST divides the triange into a triange PST and trapezium QRTS as shown in the figure. If P is 30º, find the anges Q, R, S and T of the trapezium. P 30º S T Q R 9. rectanguar fag of ength 30cms and width 20cms has three cooured vertica stripes of the same size. Find perimeter and area of each stripe. 10. fenced and has a waking path a around inside. The path is adjacent to the fence and is 1m wide. This rectanguar fence is of size 16m by 12m. Find the area of the and surrounded by the path. NSWERS 1. a) T b) F c) T d) F 2. Paraeogram 3. Hint: is a right-ange isoscees triange asic-geometry2 15 of ,

17 4. 80º 5. 22cms 6. R = 60º, S = 120º sq.cms 8. S = 105º, T = 105º, Q = 75º, R = 75º 9. 60cms, 200sq.cms sq.m asic-geometry2 16 of ,