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Geometry Developing Formulas for Triangles and Quadrilaterals

Geometry Developing Formulas for Triangles and Quadrilaterals. Warm up. Find the perimeter and area of each figure: 1) 2). x + 1. 2x. 7. x. x + 2. P = 6x + 4; A = 2x 2 + 4x 2) P = 2x + 1; A = 7x/2. Area Addition Postulate.

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Geometry Developing Formulas for Triangles and Quadrilaterals

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  1. Geometry Developing Formulas for Triangles and Quadrilaterals CONFIDENTIAL

  2. Warm up Find the perimeter and area of each figure: 1) 2) x + 1 2x 7 x x + 2 • P = 6x + 4; A = 2x2 + 4x • 2) P = 2x + 1; A = 7x/2 CONFIDENTIAL

  3. Area Addition Postulate When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces. Postulate 1:The area of a region is equal to the sum of the areas of non-overlapping parts. CONFIDENTIAL

  4. Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height. h b b CONFIDENTIAL

  5. Area: Parallelogram The area of a Parallelogram with base b and height h is A = bh. Remember that rectangles and squares are also Parallelograms. The area of a square with side s is A = s2, and perimeter is P = 4s. h b CONFIDENTIAL

  6. Finding measurements of Parallelograms Find each measurement: A) the area of a Parallelogram 5 in h 3 in Step 1: Use Pythagorean Theorem to find the height h. 6 in 32 + h2 = 52 h = 4 Step 2: Use h to find the area of parallelogram. A = bh A = 6(4) A = 24 in2 Area of a parallelogram. Substitute 6 for b and 4 for h. Simplify. CONFIDENTIAL

  7. B) the height of a rectangle in which b = 5 cm and A = (5x2 – 5x) cm2. A = bh 5x2 – 5x = 5(h) 5(x2 – x) = 5(h) x2 – x = h h = (x2 – x) cm Area of a rectangle . Substitute (5x2 – 5x) for A and 5 for b. Factor 5 out of the expression for A. Divide both sides by 5. Sym. Prop. of =. CONFIDENTIAL

  8. C) the perimeter of a rectangle in which A = 12x ft2. Step 1: Use Pythagorean Theorem to find the height h. A = bh 12x = 6(b) 2x = b Area of a rectangle . Substitute 12x for A and 6 for h. 6x Divide both sides by 6. Step 2: Use thebase and height to find the perimeter. Perimeter of a rectangle P= 2b + 2h P = 2(2x) +2(6) P = (4x +12) ft Substitute 2x for A and 6 for h. Simplify. CONFIDENTIAL

  9. Now you try! 1) Find the base of a Parallelogram in which h = 65 yd and A = 28 yd2. 1) b = 0.5 yd CONFIDENTIAL

  10. To understand the formula for the area of a triangle or trapezoid, notice that the two congruent triangles or two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or a trapezoid is half the area of the related parallelogram. h h b b b1 b2 h h b2 b2 b1 CONFIDENTIAL

  11. Area: Triangles and Trapezoids The area of a Triangle with base b and height h is A = 1 bh. 2 b1 h b The area of a Trapezoid with bases b1 and b2 and height h is A = 1 (b1 + b2)h. h 2 b2 CONFIDENTIAL

  12. Finding measurements if Triangles and Trapezoids Find each measurement: A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm. A = 1 (b1 + b2)h 2 Area of a Trapezoid. Substitute 9 for b1, 12 for b2 and 3 for h. A = 1 (9 + 12)3 2 A = 31.5 cm2 Simplify. CONFIDENTIAL

  13. B) the base of Triangle in which A = x2 in2. A = 1 bh 2 Area of a Triangle. x2 = 1 bx 2 Substitute x2 for A and x for h. x = 1 b 2 Divide both sides by x. 2x = b Multiply both sides by 2. x in b = 2x in. Sym. Prop. of =. b CONFIDENTIAL

  14. 3 ft C) b2 of the Trapezoid in which A = 8 ft2. 2 ft b2 A = 1 (b1 + b2)h 2 Area of a Trapezoid. 8 = 1 (3 +b2)2 2 Substitute 8 for A1, 3 for b1 and 2 for h. 8 = 3 +b2 Multiply ½ by 2. 5 = b2 Subtract 3 from both sides. b2 = 5 ft. Sym. Prop. of =. CONFIDENTIAL

  15. Now you try! 2) Find the area of the triangle. 20 m 12 m b 2) b = 96 m2 CONFIDENTIAL

  16. A kite or a rhombus with diagonald1 and d2 can be divided into two congruent triangles with a base d1 and height of ½ d2 . area of each triangle: A = 1 d1(½ d2) 2 =1 d1d2 4 ½ d2 d1 Total area : A =2(1 d1d2 ) =1 d1d2 42 ½ d2 d1 CONFIDENTIAL

  17. Area: Rhombus and kites The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 . 2 ½ d2 ½ d2 d1 d1 CONFIDENTIAL

  18. Finding measurements of Rhombus and kites Find each measurement: A) d2 of a kite with d1 = 16 cm, and A = 48 cm2. A = 1 (d1d2) 2 Area of a kite. Substitute 48 for A, 16 for d1. 48 = 1 (16)d2 2 d2 = 6 cm Simplify. CONFIDENTIAL

  19. d2 = (10x + 10)in. B) The area of the rhombus . d1 = (6x + 4)in. A = 1 (d1d2) 2 Area of a kite. Substitute (6x + 4) for d1 and (10x + 10) for d2. A = 1 (6x + 4) (10x + 10) 2 A = 1 (6x2 + 100x + 40) 2 Multiply the binomials. A = (3x2 + 50x + 20) Simplify. CONFIDENTIAL

  20. C) The area of the kite. 41 ft 9 ft y x 15 ft Step 1: The diagonald1 and d2 form four right angles. Use Pythagorean Theorem to find the x and y. 92 + x2 = 412 x2 = 1600 x = 40 92 + y2 = 152 y2 = 144 y = 12 CONFIDENTIAL

  21. 41 ft 9 ft Step 2: Use d1 and d2 to find the area. d1 = (x + y) which is 52. Half of d2 = 9, so d2 = 18. y x 15 ft A = 1 (d1d2) 2 Area of a kite. Substitute 52 for d1 and 18 for d2. A = 1 (52) (18) 2 A = 468 ft2 Simplify. CONFIDENTIAL

  22. Now you try! 3) Find d2 of a rhombus with d1 = 3x m, and A = 12xy m2. 3) b = 96 m2 CONFIDENTIAL

  23. Games Application The pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the red square. Perimeter: Each side of the red square is the diagonal of the square grid. Each grid square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the red square is P = 4s = 4 √2 cm. CONFIDENTIAL

  24. Area: Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2. A = 1 (d1d2) = 1 (√2)(√2) = 2 cm2. 22 Method 2: Theside length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2. CONFIDENTIAL

  25. Now you try! 4) Find the area and perimeter of the large yellow triangle in the figure given below. 4) A = 4 cm2 P = 4 + 4√2 cm CONFIDENTIAL

  26. Now some problems for you to practice ! CONFIDENTIAL

  27. Assessment Find each measurement: 1) the area of the Parallelogram. 2) the height of the rectangle in which A = 10x2 ft2. 10 cm 2x ft 12 cm • 120 cm2 • 2) 5x ft CONFIDENTIAL

  28. Find each measurement: 20 m 9 m 3) the area of the Trapezoid. 15 m 9 in 4) the base of the triangle in which A = 58.5 in2. 3) 240 m2 4) 13 in CONFIDENTIAL

  29. Find each measurement: 5) the area of the rhombus. 25 in 15 m 6) d2 of the kite in which A = 187.5 m2. 14 in 5) 175 in2 6) 25 m CONFIDENTIAL

  30. 7) The rectangle with perimeter of (26x + 16) cm and an area of (42x2 + 51x + 15) cm2. Find the dimensions of the rectangle in terms of x. 7) (7x + 5) and (6x + 3) CONFIDENTIAL

  31. 8) The stained-glass window shown ii a rectangle with a base of 4 ft and a height of 3 ft. Use the grid to find the area of each piece. 8) √10 ft2 CONFIDENTIAL

  32. Let’s review When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces. Postulate 1:The area of a region is equal to the sum of the areas of non-overlapping parts. CONFIDENTIAL

  33. Area: Parallelogram The area of a Parallelogram with base b and height h is A = bh. Remember that rectangles and squares are also Parallelograms. The area of a square with side s is A = s2, and perimeter is P = 4s. h b CONFIDENTIAL

  34. Finding measurements of Parallelograms Find each measurement: A) the area of a Parallelogram 5 in h 3 in Step 1: Use Pythagorean Theorem to find the height h. 6 in 32 + h2 = 52 h = 4 Step 2: Use h to find the area of parallelogram. A = bh A = 6(4) A = 24 in2 Area of a parallelogram. Substitute 6 for b and 4 for h. Simplify. CONFIDENTIAL

  35. Area: Triangles and Trapezoids The area of a Triangle with base b and height h is A = 1 bh. 2 b1 h b The area of a Trapezoid with bases b1 and b2 and height h is A = 1 (b1 + b2)h. h 2 b2 CONFIDENTIAL

  36. Finding measurements if Triangles and Trapezoids Find each measurement: A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm. A = 1 (b1 + b2)h 2 Area of a Trapezoid. Substitute 9 for b1, 12 for b2 and 3 for h. A = 1 (9 + 12)3 2 A = 31.5 cm2 Simplify. CONFIDENTIAL

  37. A kite or a rhombus with diagonald1 and d2 can be divided into two congruent triangles with a base d1 and height of ½ d2 . area of each triangle: A = 1 d1(½ d2) 2 =1 d1d2 4 ½ d2 d1 Total area : A =2(1 d1d2 ) =1 d1d2 42 ½ d2 d1 CONFIDENTIAL

  38. Area: Rhombus and kites The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 . 2 ½ d2 ½ d2 d1 d1 CONFIDENTIAL

  39. Finding measurements of Rhombus and kites Find each measurement: A) d2 of a kite with d1 = 16 cm, and A = 48 cm2. A = 1 (d1d2) 2 Area of a kite. Substitute 48 for A, 16 for d1. 48 = 1 (16)d2 2 d2 = 6 cm Simplify. CONFIDENTIAL

  40. Games Application The pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the red square. Perimeter: Each side of the red square is the diagonal of the square grid. Each grid square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the red square is P = 4s = 4 √2 cm. CONFIDENTIAL

  41. Area: Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2. A = 1 (d1d2) = 1 (√2)(√2) = 2 cm2. 22 Method 2: Theside length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2. CONFIDENTIAL

  42. You did a great job today! CONFIDENTIAL

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