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Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
Geometry Section 6-4 1112
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Geometry Section 6-4 1112

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Rectangles

Rectangles

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  • 1. SECTION 6-4 Rectangles Tuesday, April 29, 14
  • 2. ESSENTIAL QUESTIONS How do you recognize and apply properties of rectangles? How do you determine if parallelograms are rectangles? Tuesday, April 29, 14
  • 3. RECTANGLE Tuesday, April 29, 14
  • 4. RECTANGLE A parallelogram with four right angles. Tuesday, April 29, 14
  • 5. RECTANGLE A parallelogram with four right angles. Four right angles Tuesday, April 29, 14
  • 6. RECTANGLE A parallelogram with four right angles. Four right angles Opposite sides are parallel and congruent Tuesday, April 29, 14
  • 7. RECTANGLE A parallelogram with four right angles. Four right angles Opposite sides are parallel and congruent Opposite angles are congruent Tuesday, April 29, 14
  • 8. RECTANGLE A parallelogram with four right angles. Four right angles Opposite sides are parallel and congruent Opposite angles are congruent Consecutive angles are supplementary Tuesday, April 29, 14
  • 9. RECTANGLE A parallelogram with four right angles. Four right angles Opposite sides are parallel and congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other Tuesday, April 29, 14
  • 10. THEOREMS 6.13 - Diagonals of a Rectangle: 6.14 - Diagonals of a Rectangle Converse: Tuesday, April 29, 14
  • 11. THEOREMS 6.13 - Diagonals of a Rectangle: If a parallelogram is a rectangle, then its diagonals are congruent 6.14 - Diagonals of a Rectangle Converse: Tuesday, April 29, 14
  • 12. THEOREMS 6.13 - Diagonals of a Rectangle: If a parallelogram is a rectangle, then its diagonals are congruent 6.14 - Diagonals of a Rectangle Converse: If diagonals of a parallelogram are congruent, then the parallelogram is a rectangle Tuesday, April 29, 14
  • 13. EXAMPLE 1 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and LN = 6.5 feet, find KM. Tuesday, April 29, 14
  • 14. EXAMPLE 1 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and LN = 6.5 feet, find KM. Since we have a rectangle, the diagonals are congruent. Tuesday, April 29, 14
  • 15. EXAMPLE 1 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and LN = 6.5 feet, find KM. Since we have a rectangle, the diagonals are congruent. The diagonals also bisect each other, so JN = LN and KN = MN. Tuesday, April 29, 14
  • 16. EXAMPLE 1 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and LN = 6.5 feet, find KM. Since we have a rectangle, the diagonals are congruent. The diagonals also bisect each other, so JN = LN and KN = MN. So JN = LN = KN = MN = 6.5 feet and KM = KN + MN. Tuesday, April 29, 14
  • 17. EXAMPLE 1 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet and LN = 6.5 feet, find KM. Since we have a rectangle, the diagonals are congruent. The diagonals also bisect each other, so JN = LN and KN = MN. So JN = LN = KN = MN = 6.5 feet and KM = KN + MN. KM = 13 feet Tuesday, April 29, 14
  • 18. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. Tuesday, April 29, 14
  • 19. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 Tuesday, April 29, 14
  • 20. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 Tuesday, April 29, 14
  • 21. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 11x + 2 = 90 Tuesday, April 29, 14
  • 22. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 11x + 2 = 90 −2 −2 Tuesday, April 29, 14
  • 23. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 11x + 2 = 90 −2 −2 11x = 88 Tuesday, April 29, 14
  • 24. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 11x + 2 = 90 −2 −2 11x = 88 11 11 Tuesday, April 29, 14
  • 25. EXAMPLE 2 Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)° and m∠SUR = (3x − 2)°, find x. m∠RTU + m∠SUR = 90 8x + 4 + 3x − 2 = 90 11x + 2 = 90 −2 −2 11x = 88 11 11 x = 8 Tuesday, April 29, 14
  • 26. EXAMPLE 3 Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides of the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, and DA = 35 inches, how long do the lengths of the diagonals need to be? Tuesday, April 29, 14
  • 27. EXAMPLE 3 Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides of the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, and DA = 35 inches, how long do the lengths of the diagonals need to be? The diagonal forms a right triangle with legs of 12 and 35. We need to find the hypotenuse. Tuesday, April 29, 14
  • 28. EXAMPLE 3 Tuesday, April 29, 14
  • 29. EXAMPLE 3 a2 + b2 = c2 Tuesday, April 29, 14
  • 30. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 Tuesday, April 29, 14
  • 31. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 144 + 1225 = c2 Tuesday, April 29, 14
  • 32. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 144 + 1225 = c2 1369 = c2 Tuesday, April 29, 14
  • 33. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 144 + 1225 = c2 1369 = c2 1369 = c2 Tuesday, April 29, 14
  • 34. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 144 + 1225 = c2 1369 = c2 1369 = c2 c = 37 Tuesday, April 29, 14
  • 35. EXAMPLE 3 a2 + b2 = c2 122 + 352 = c2 144 + 1225 = c2 1369 = c2 1369 = c2 c = 37 The diagonals must both be 37 inches Tuesday, April 29, 14
  • 36. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. Tuesday, April 29, 14
  • 37. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. Diagonals must be congruent Tuesday, April 29, 14
  • 38. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 Diagonals must be congruent Tuesday, April 29, 14
  • 39. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 Diagonals must be congruent Tuesday, April 29, 14
  • 40. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 Diagonals must be congruent Tuesday, April 29, 14
  • 41. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 Diagonals must be congruent Tuesday, April 29, 14
  • 42. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 Diagonals must be congruent Tuesday, April 29, 14
  • 43. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 Diagonals must be congruent Tuesday, April 29, 14
  • 44. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 Diagonals must be congruent Tuesday, April 29, 14
  • 45. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 Diagonals must be congruent Tuesday, April 29, 14
  • 46. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 47. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 48. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 49. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 50. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 51. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 52. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 53. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 = −6 2 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 54. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 = −6 2 = −3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 55. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 = −6 2 m( JM) = −3 − 3 0 + 2 = −3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 56. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 = −6 2 m( JM) = −3 − 3 0 + 2 = −6 2 = −3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 57. EXAMPLE 4 Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2), and M(0, −3). Determine whether JKLM is a rectangle by using the distance formula, then slope. JL = (−2 − 3)2 + (3 + 2)2 = (−5)2 + 52 = 25 + 25 = 50 KM = (1 − 0)2 + (4 + 3)2 = 12 + 72 = 1 + 49 = 50 m( JK) = 4 − 3 1 + 2 = 1 3 m(LM) = −3 + 2 0 − 3 = −1 −3 = 1 3 m(KL) = −2 − 4 3 − 1 = −6 2 m( JM) = −3 − 3 0 + 2 = −6 2 = −3 = −3 Diagonals must be congruent Opposite sides parallel, consecutive sides perpendicular Tuesday, April 29, 14
  • 58. PROBLEM SET Tuesday, April 29, 14
  • 59. PROBLEM SET p. 422 #1-31 odd, 41, 49, 55, 59, 61 “Character - the willingness to accept responsibility for one's own life - is the source from which self respect springs.” - Joan Didion Tuesday, April 29, 14

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