CHAPTER VII - POLYGONS

1. CHAPTER VII - POLYGONS Maths – IX

2. Module Objectives • DefinePolygon • Identify and name Polygons • Differentiatebetweenregular and irregularpolygons • Inscriberegularpentegon, hexagon and octagon.

3. Introduction The common properties in the figures are mentioned below. They are closed figures. They are bounded by three or more line segments. The line segments are co-planar and non-collinear. The line segments intersect each other at their end points. Each end point is shared only by two line segments. Such geometrical figures are called Polygons.

4. Polygons • .A polygonis a figure enclosed by three or more line segments which are co-planar, non-collinear and intersectingeachotherattheir end points. • The end points of the line segments are calledvertices of the Polygon. The line segments whichmake a polygon are calledsides of the polygon.

5. Polygons • In a polygon the number of sides are equal t o the number of angles. • The polygons are namedaccording to the number of sidestheycontain: Study the following table.

6. Polygons .

7. Diagonals of a Polygon A B F A line segment joininganytwo non consecutivevertices of a polygoniscalledits diagonal. In the adjoining figure AC, AD, AE are the diagonals. Draw the other possible diagonals in the figure fromB,C,D,E and F. How many diagonals canbe drawn in a hexagonABCDEF? C E D

8. Regular Polygon 90° Observe the polygons drawn below.. Compare the magnitude of sides and the angles in each case. 90° 90° 90° 120° 108° 120° 120° 108° 108° 120° 120° 108° 108° 120° Wefindthat in each case all the sides are equal to eachother and all the angles are equal to eachother. They are calledregularpolygon. A polygonthatisbothequilateral and equiangulariscalled a regularpolygon.

9. Interiorregion of a Polygon A Interior Region Observe the adjoining figure. The polygon ABC or ▲ABC encloses some part of the plane which is represented by the shaded part. This is called the interior region of the polygon. Hence the area of the polygon is the polygonal region formed by the union of the polygon and its interior region. Area of a triangle is equal to the triangular region formed by the union of the triangle and its interior. Similarly, Area of a quadrilateral is equal to the quadrilateral region formed by the union of the quadrilateral and its interior. P Interior Region B C S Q R

10. Inscribing Regular Polygons A polygon inscribed in a circle is a polygon whose vertices lie on the circle. Observe the adjacent figure. The polygon ABCDE is inscribed in a circle. What are the sides, AB,BC,CD.DE.EA of the polygon with respect to the circle? They are chords of the circle. We know that in a regular pentagon all the sides are equal. Observe the adjoining figure, where AB,BC,CD,DE,EA are equal. D A E E A D D B C B C

11. E Inscribing Regular Polygons A D B Now let us draw the central angles subtended by each of thesechords. Observe the number of angles formedat the centre of the circle. Fromthisactivityweshallconclude the following. The number of angles subtended by the sides of an inscribedpolygonat the centre isequal to the number of sides of the polygon. All the angles formedat the centre by the sides of the inscribedpolygon are equal. We know that the angle at the centre of the circleis 360°. C

12. D Inscribing Regular Polygons E C A If a regularpolygonwith ‘n’ sidesisinscribedinside a circle , each central angle subtended by itssidesisequal to . In a pentegoneach central angle = To inscribe a regularpentegoninside a circle of radius 3cm, Step 1: Calculate the angle at the centre of the pentagon. Step2: With O as centre draw a circle of radius 3cm. Step3: Construct the central angle, Step4: With AB are radius and B as centre step off equal arc on the circle to get the point C. Similarly get the point D and E. Step5: Join Ab,Bc,CD,DE,EA. ABCDE is a pentegoninsribed inside a circle. B O

13. Inscribing Regular Polygons D E C F B To inscribe a regularHexagoninside a circle of radius 4cm, Step 1: Calculate the angle at the centre of the hexagon. central angle = Step2: With O as centre draw a circle of radius 4cm. Step3: Construct the central angle, Step4: With AB are radius and B as centre step off equal arc on the circle to get the point C. Similarly get the point D,E and F. Step5: Join AB,BC,CD,DE,EF,FA. ABCDEF is a hexagon inscribed inside a circle. A O

14. Alternatemethod to inscribehexagon in a circle In the aboveregularhexagon, what type of triangle isOAB? In ▲OAB, OA= OB (radii of the samecircle) each () ▲OABis an equilateral Triangle. This propertycanbeused to inscribe a regularhexagon. Step1: Mark any point on the circlesay A, and step off arcs equal to the radius of the givencircle, on the circle. Let the interceptsbeB,C,D,E and F Step2: Join AB,BC,CD,DE,EF,FA. ABCDEF is a hexagon inscribed inside a circle.

15. Inscribing Regular Polygons C B D A To inscribe a regularOctagoninside a circle of radius 4cm, Step 1: Calculate the angle at the centre of the hexagon. central angle = Step2: With O as centre draw a circle of radius 4cm. Step3: Construct the central angle, Step4: With AB are radius and B as centre step off equal arc on the circle to get the point C. Similarly get the point D,E,F,G,H. Step5: Join AB,BC,CD,… ABCDEFGHis a octagon inscribed inside a circle. O H E G F

16. Exercises