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Understanding Areas of Trapezoids: Formulas and Exercises

This guide explores the area of trapezoids, highlighting key concepts such as bases, altitude, and median. The trapezoid's area can be calculated using the formula A = ½h(b₁ + b₂), where h is the height and b₁, b₂ are the lengths of the bases. Detailed examples illustrate this formula and its application in various scenarios, including isosceles trapezoids. Practice exercises are included to reinforce understanding, making it an essential resource for geometry learners.

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Understanding Areas of Trapezoids: Formulas and Exercises

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  1. Geometry 11.3 Areas of Trapezoids

  2. Trapezoids In a trapezoid, the bases are the parallel sides. An altitude of a trapezoid is defined in the same way as an altitude of a parallelogram. The altitude of a trapezoid is any segment perpendicular to a line containing one base from a point on the opposite base. In a trapezoid, all altitudes have the same length, called the height, h. b1 h h h b2

  3. Area of a Trapezoid The area of a trapezoid equals half the product of height and the sum of the bases. b1 8 A = ½h(b1+ b2) 6 h or 12 A = mh b2 m is the median, the average of the bases Area = ½(6)(8 + 12) = 60 square units Area = 10(6) = 60 square units

  4. 10 15 45 18 12 60 7 11 3 30 10 8 Exercises Try # 2 and #4! 30º 8 2 6√3 8 6 2 1. A = ½(11)(18+8) 3. A = ½(6√3)(30+15) A = 143 A = 135√3 2. A = ½(10)(7+3) 4. A = ½(8)(10+2) A = 50 A = 48

  5. Easy method : Find m first A = ½h(b1+ b2) m is the median length of the trapezoid m = ½ (b1+ b2) So, more simply: It is the average of the bases A = h•m

  6. Exercises 5 3 15 6x 8 = h•m 70 35 edian 14 17.5 11.5 5.5 10x 7. 46 = 4m m = 11.5 11.5 = ½(8 + b2) 23 = 8 + b2 b2 = 15 9. 6√3m = 33√3 m = 5.5 5.5 = ½(b1 + 8) 11 = b1 + 8 b1 = 3 5. m = ½(15 + 13) m = 14 A = 5(14) = 70 6. m = ½(25 + 10) m = 17.5 140 = 17.5 h h = 8 10. 70x² = 7x • m m = 10x b1 = 6x 8. 7 = ½(b1 + 9) 14 = b1 + 9 b1 = 5

  7. 24 10 60 60 26 12 26 12 12 30 Exercises Find the area of each isosceles trapezoid. 5-12-13 Triple 6 6√3 x 10 10 m = ½(24 + 12) x² + 10² = 26² m = ½(30+ 10) m = 18 x² + 100 = 676 m = 20 x² = 576 A = 6√3 • 18 A = 24 • 20 x = √576 A = 108√3 A = 480 x = 24

  8. Exercises 13. Find the area of an isosceles trapezoid with legs 25 cm and bases 16 cm and 30 cm. 16 7-24-25 Triple 25 25 24 30 - 16 7 30 7 2 m = ½(16 + 30) 552 cm² m = 23 A = 24 • 23 = 552

  9. Exercises 14. Find the area of a trapezoid with 45˚ base angles and bases 17 and 23. 45-45-90 Rt. ∆ 17 3√2 3 45 45 23 - 17 3 23 3 2 m = ½(17 + 23) m = 20 A = 3 • 20 = 60 60 sq. units

  10. Homework pg. 436 #1-19, 23 odd pg. 470 #1-9 For #5 see the bonus from Powerpoint 11.2

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