1 / 25

Quadrilateral

Quadrilateral. 45 0. 2 pairs of equal adjacent sides. 1 pair of // opp. Sides. Kite. Trapezium. Sum of interior angles is 180 0. One of the diagionals is axis of symmetry. Parallelogram. 2 diagionals are . 2 pairs of opp.// sides. 4 equal sides. Rhombus. Properties of trapesium.

margaux
Download Presentation

Quadrilateral

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Quadrilateral 450 2 pairs of equal adjacent sides 1 pair of // opp. Sides Kite Trapezium Sum of interior angles is 1800 One of the diagionals is axis of symmetry Parallelogram 2 diagionals are  2 pairs of opp.// sides 4 equal sides Rhombus Properties of trapesium 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal (opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Properties of // gram and kite Diagonals bisects each interior angle 4 right angles Rectangle 4 right angles and 4 equal sides Properties of // gram Properties of rhombus/rectangle Diagonals are equal Angles between each diagional and each side is 450 Square

  2. Trapeziums Sum of interior angles is 1800 Definition : 1 pair of parallel sides Properties:

  3. Parallelogram 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Definition : 2 pairs of opp. parallel sides Properties:

  4. Conditions for Parallelogram If 2 pairs of opposite sides are equal thenthe quadrilateral is parallelogram.(opp. sides eq.) If 1 pair of opposite sides is equal and parallel thenthe quadrilateral is parallelogram (opp. sides eq. and //) If diagonals bisect each other thenthe quadrilateral is parallelogram (diag. Bisect each other) If 2 pairs of opposite angles are equal then the quadrilateral is parallelogram.(opp. s of eq.)

  5. Rhombus Definition : a // gram or a kite of 4 equal sides 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisects each interior angle Diagonals are  Diagonals bisect each other (diag. Of // gram) Properties:

  6. Rectangle 2 pairs of opposite sides are equal.(opp. sides of // gram) 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) Diagonals are equal Definition : a parallelogram of 4 right angles Properties:

  7. Square Definition : a // gram of 4 right angles and 4 equal sides 2 pairs of opposite angles are equal(opp. s of // gram) Diagonals bisect each other (diag. Of // gram) 450 Diagonals are equal Diagonals are  Angles between each diagonal and each side is 450 Properties: 2 pairs of opposite sides are equal.(opp. sides of // gram)

  8. Example 1: In the figure, PQRS is a kite P y+3 x+1 S Q x+y 8 R • Find x and y. • Find the perimeter of the kite PQRS PQ = PS (given) x+1 = y+3x-y=2 (1) QR=SR (given)x+y=8 (2) (1)+(2), 2x=10 x=5Put x=5 into (1), 5-y=2 y=3 (a) (b) PQ = x+1=5+1=6 PQ+PS+SR+QR = 6 + 6 + 8 + 8 =28

  9. Example 2: In the figure, ABCD is a kite. E is a point of intersection of diagonals AC and BD, AE=9 cm, EC=16 cm and DE=EB=12 cm D 12 16 C A 9 E 12 B • Find the area of ABCD. • Find the perimeter of ABCD • ABC= ADC (axis of symmetry AC)AED=900 Area of ADC = Area of kite ABCD=Area of ABC+Area of ADC = 150+150 =300 cm2 In ADE, AD2=AE2+DE2=92+122=225 cm2 (Pyth theorem) AD=15 cmIn CDE, DC2=DE2+EC2=122+162=400 cm2 (Pyth theorem) DC=20 cm  Perimeter of ABCD=AD+AB+ DC+CB = 15 + 15 + 20 + 20 =70 cm (b)

  10. Example 3: In the figure, ABCD is a parallelogram. Find x and y. A D x 1500-y 2y 680 B C AD//BC (Given) x+680=1800(prop. Of trapezium)  x=1120(1500-y)+2y=1800(prop. Of trapezium) 1500+y=1800  y=1800 -1500=300

  11. Example 4: In the figure, ABCD is a parallelogram. Find x and y. D C 3x+100 x+200 y A B DAB=DCB (opp. s of // gram) x+200=3x-100 2x=300x=150DAB+CBA=1800(int.s , AD//BC) x+200+y=1800 150+200+y=1800y=1450

  12. Example 5: In the figure, ABCD is a isosceles trapezium with AB=DC.Find x , y and z A D z 1260 a Construct AE // DC  AD//EC and AE//DC ADCE is a parallelogram (Definition of // gram) E x y  B C  ADCE is a parallelogram (proof)AE=DC (opp.sides of // gram)  AD//BC (Given) x+1260=1800(prop. Of trapezium)  x=540 In ABE,AE=DC (proof) AB=AC (given) AB=AE  y=a (base s. isos )a= x(corr.s. AE//DC)  y=x =540 y+z=1800(prop. Of trapesium)z= 1800-540 = 1260

  13. MID-POINT THEOREM A N M C B IF AM = MB and AN =NC then (a) MN // BC (b) MN = (Abbreviation: Mid-point theorem)

  14. Example 13: In the figure, ABC is a triangle, find x and y. C y E 6 420 x B A D CE=BE (given) AD=DB (given) (mid-point theorem) DE//AC x = EDB =420 (corr. s , DE//AC) (mid-point theorem)

  15. Example 14: Prove that BPQR is a parallelgram A R Q C B P (given) AR=RB (given) AQ=QC (mid-point theorem) (given) (opp-sides eq. And //)

  16. Ex 11D D A M y cm x cm 5 cm B N C 1(b) BM=MD (given) BN=NC (given) AM=AC (given) BN=NC (given) (mid-point theorem) (mid-point theorem)

  17. Ex 11D A 1100 a Q P 460 C B 2(b) AP=BP (given) AQ=CQ (given) (mid-point theorem) (corr.s. PQ//BC) In APQ, APQ+ PAQ+ a = 1800 460+1100+a=1800 a=240 (adj s. on a st line)

  18. A 10 8 F E A B C D 9 60 F E 70 50 3(a) 3(b) B C D 9

  19. 4. BP=PA (given) CR=RB (given) C (mid-point theorem) P R AQ=QB (given) AP=PC (given) B A Q Area of ABC  (mid-point theorem) 6 8

  20. INTERCEPT THEOREM transversal X intercept Y B P A D C Q

  21. INTERCEPT THEOREM If AB//CD//EF then (intercept theorem) A B D C F E

  22. INTERCEPT THEOREM Construct GB through A such that BG//CD//EF (given) GB//CD//EF B G (intercept theorem) Proved: A D C E F

  23. Example 15. AP//BQ//CR, AB=BC, AP=11 and CR=5. Find BQ. Join AR to cut BQ at S (given) AP//BQ//CR (given) S (intercept theorem) (given) (proved) (proved) (proved) (mid-pt theorem) (mid-pt theorem) A B C 11 5 P Q R BQ=BS+SQ = 2.5+5.5=8

  24. Example 16. AB and DC are straight lined. Find x and y. (b) AB=6, PB=2 and AQ=9. Find QC (proved) (a) Proved: A E D Join DE through A and // BC DE//PQ//BC (given) Q P (intercept theorem) B C

  25. Example 16. Find QR and CD. AP//BQ//CR (given) BQ//CR//DS (given) (intercept theorem) (intercept theorem) A P 3 2 Q B 6 R C 8 S D

More Related