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Exploring Area Calculations: Right Triangles, Parallelograms, and Beyond

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This resource focuses on developing and applying formulas for calculating areas of various geometrical shapes, including triangles, parallelograms, and trapezoids. It leverages principles such as the Pythagorean Theorem to find unknown dimensions and areas effectively. Engaging exercises such as solving tangrams and applying the Area Addition Postulate enhance understanding. With clear examples and practice problems, this material reinforces key mathematical concepts and encourages critical thinking in geometry.

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Exploring Area Calculations: Right Triangles, Parallelograms, and Beyond

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Presentation Transcript

  1. Do Now 05/02/2014 Find the unknown side length in each right triangle with legs a and b and hypotenuse c. 1.a = 20, b = 21 2.b = 21, c = 35

  2. 9.1 Developing Formulas for Triangles and Quadrilaterals Objective: Develop and apply the formulas for the areas of triangles and special quadrilaterals.

  3. Let’s Talk Common Sense Area

  4. Tangram 7 shapes that fit together to make a square…yet it turned into a type of art.

  5. Solve a Tangram A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. http://pbskids.org/cyberchase/math-games/tanagram-game/ FOX How Would we find the area of the whole Fox?

  6. Area Addition Postulate The area of a region is equal to the sum of the areas of its non- overlapping parts

  7. Recall that a rectangle with base b and height h has an area of A = bh.

  8. Area of a Parallelogram The area of a parallelogram with base b and height h is Area=bh b= base (long side) H= height (vertical,not slanted) *Remember that rectangles and squares are also parallelograms.

  9. Pythagorean Theorem • If you have a right triangle, a2 + b2 = c2 where c is the hypotenuse

  10. Example 1: Find the area

  11. Example 2: Find the area of the parallelogram Step 1: Use the Pythagorean Theorem to find the height h. Step 2 Use h to find the area of the parallelogram

  12. Example 3 Find the height of a rectangle in which b = 3 in. and A = (6x² + 24x – 6) in2.

  13. Example 4 • Find the perimeter of the rectangle, in which A = (79.8x2 – 42) cm2 Step 1 Use the area and the base to find the height. Step 2 Use the base and the height to find the perimeter.

  14. Check For Understanding Find each measurement. • 1. the height of the parallelogram, in which A = 182x2 mm2 2. the perimeter of a rectangle in which h = 8 in. and A = 28x in2

  15. Check For understanding 3) Find the base of the parallelogram in which h = 56 yd and A = 28 yd2. 4) Find the area of a square with one side length of 4x-2 in2.

  16. Area of Triangles and Trapezoids • The area of a triangle or trapezoid is half the area of the related parallelogram.

  17. Area: Triangles and Trapezoids

  18. Example 1 • Find the area of a trapezoid in which b1 = 8 in., b2 = 5 in., and h = 6.2 in. b) Find the area of a triangle where b= 2x and h= 25

  19. Example 2 • Find the base of the triangle, in which A = (15x2) cm2, when h= 10x cm.

  20. Example 3 • Find b2 of the trapezoid, in which A = 231 mm2.

  21. Check For understanding 1) Find the area of the triangle. ( use Pythagorean THM. to find b 1st)

  22. Check For understanding 2) Find the area of the trapezoid • 3) Find the base of a triangle in which h = 8 cm and A = (12x + 8) cm2

  23. 10 minute Break

  24. Recall The diagonals of a rhombus and kite are perpendicular, and the diagonals of a rhombus bisect each other.

  25. Area: Rhombi and Kites A= ½ d1d2

  26. Example 1 Find d2 of a kite in which d1 = 14 in. and A = 238 in2.

  27. Example 2 • Find the area of a rhombus.

  28. Example 3 Find the area of the kite • Step 1 The diagonals d1 and d2 form four right triangles. Use the Pythagorean Theorem to find x.

  29. Example 3 Continued • Step 2 Use d1 and d2 to find the area. d1 =48 , d2 =42.

  30. Warm – Up 1)Identify the base and height of the parallelogram b= h= 2)Find the area of the parallelogram 12 x in. 5x+4 in.

  31. Check For Understanding

  32. Check For Understanding 3) Find the d2 of the rhombus. 4) Find d2 of a rhombus in which d1 = 3x m and A = 12xy m2.

  33. Homework Practice A 1-11( No Variables ! )

  34. Make your own formula sheet

  35. Do Now 10x )

  36. Check Homework

  37. Self Quiz 60 in

  38. Bell Ringer State the area formulas for quadrilaterals and triangles in the grid on your worksheet.

  39. Angry Area

  40. Rules -Each question is worth 100 points -When your team has an answer, you are allowed to ring in once. • If you are incorrect, you lose 50 points and the remaining teams have 1 minute to answer. (keep working) - team with most points at the end wins!

  41. 1)Find each measurement The perimeter of the rectangle: ( 2x+2) in. ( x-1) in.

  42. 2)Find each measurement The area of a parallelogram in which b= (x+5)ftand h= (2x-1) ft.

  43. 3)Find each measurement 2m

  44. 4)Find each measurement The area of the whole triangle 25 in. 17 in. 15 in. 8in.

  45. 5) Find the measurement 10 cm. 4cm.

  46. 6) Find the measurement The area of a trapezoid : 15 m. 20 m. 9 m.

  47. 7) Find the measurement The area of a triangle in which b= (x+1) ft and h= 8x ft

  48. 8) Find the measurement 2x ft.

  49. 9) Find the measurement 6xft.

  50. 10 ) Find the measurement

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