MATHS PROJECT Quadrilaterals

1. MATHS PROJECTQuadrilaterals -Monica Sant IX-A

2. Definition • A plane figure bounded by four line segments AB,BC,CD and DA is called a quadrilateral. A B C D

3. In geometry, a quadrilateral is a polygon with four sides and four vertices. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistence with pentagon. There are over 9,000,000 quadrilaterals. Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave.

4. Taxonomic Classification The taxonomic classification of quadrilaterals is illustrated by the following graph.

5. Types of Quadrilaterals • Parallelogram • Trapezium • Kite

7. Parallelogram

8. Types of Parallelograms

10. Is a square a rectangle? Some people define categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square. This is appropriate for everyday use of the words, as people typically use the less specific word only when the more specific word will not do. Generally a rectangle which isn't a square is an oblong. But in mathematics, it is important to define categories inclusively, so that a square is a rectangle. Inclusive categories make statements of theorems shorter, by eliminating the need for tedious listing of cases. For example, the visual proof that vector addition is commutative is known as the "parallelogram diagram". If categories were exclusive it would have to be known as the "parallelogram (or rectangle or rhombus or square) diagram"!

11. Trapezium I have only one set of parallel sides. [The medianof a trapezium is parallel to the bases and equal to one-half the sum of the bases.]

12. Kite Ithas two pairs of sides. Each pair is made up of adjacent sides (the sides meet) that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

13. Some other types of quadrilaterals Cyclic quadrilateral: the four vertices lie on a circumscribed circle. Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible. Bicentric quadrilateral: both cyclic and tangential.

14. Angle Sum Property Of Quadrilateral The sum of all four angles of a quadrilateral is 360.. A D 1 6 5 2 4 3 B C Given: ABCD is a quadrilateral To Prove: Angle (A+B+C+D) =360. Construction: Join diagonal BD

15. Proof: In ABD Angle (1+2+6)=180 - (1) (angle sum property of ) In BCD Similarly angle (3+4+5)=180 – (2) Adding (1) and (2) Angle(1+2+6+3+4+5)=180+180=360 Thus, Angle (A+B+C+D)= 360

16. The Mid-Point Theorem The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it. A 3 1 E D F 2 4 B C Given: In ABC. D and E are the mid-points of AB and AC respectively and DE is joined To prove: DE is parallel to BC and DE=1/2 BC

17. Construction: Extend DE to F such that De=EF and join CF Proof: In AED and CEF Angle 1 = Angle 2 (vertically opp angles) AE = EC (given) DE = EF (by construction) Thus, By SAS congruence condition AED= CEF AD=CF (C.P.C.T) And Angle 3 = Angle 4 (C.P.C.T) But they are alternate Interior angles for lines AB and CF Thus, AB parallel to CF or DB parallel to FC-(1) AD=CF (proved) Also AD=DB (given) Thus, DB=FC -(2) From (1) and(2) DBCF is a gm Thus, the other pair DF is parallel to BC and DF=BC (By construction E is the mid-pt of DF) Thus, DE=1/2 BC

18. THE END- MONICA SANTIX-AROLL NO. 31