I've been trying to attempt this in many different ways but I can't seem to get it right or figure out how Id go about calculating the area.?

Can someone please explain how I'd go about solving this pleaseeee

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3 Answers

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    Probably the easiest way to do this is to find the area of the large rectangle and then subtract the areas of the 3 unshaded triangles

    12 + 4 = 16

    6 + 2 = 8

    8 * 16 = 128

    (1/2) * 4 * 6 = 12

    (1/2) * 12 * 8 = 48

    (1/2) * 2 * 16 = 16

    12 + 48 + 16 = 76

    128 - 76 = 52

    The shaded region has an area of 52 square cm.

    Another way involves the Pythagorean Theorem and Heron's formula

    4^2 + 6^2 = a^2

    16 + 36 = a^2

    52 = a^2

    2 * sqrt(13) = a

    12^2 + 8^2 = b^2

    144 + 64 = b^2

    208 = b^2

    4 * sqrt(13) = b

    2^2 + 16^2 = c^2

    4 + 256 = c^2

    4 * 65 = c^2

    2 * sqrt(65) = c

    s = (a + b + c) / 2

    s = (2 * sqrt(13) + 4 * sqrt(13) + 2 * sqrt(65)) / 2

    s = (6 * sqrt(13) + 2 * sqrt(65)) / 2

    s = 3 * sqrt(13) + sqrt(65)

    A^2 = s * (s - a) * (s - b) * (s - c)

    A^2 = (3 * sqrt(13) + sqrt(65)) * (3 * sqrt(13) + sqrt(65) - 2 * sqrt(13)) * (3 * sqrt(13) + sqrt(65) - 4 * sqrt(13)) * (3 * sqrt(13) + sqrt(65) - 2 * sqrt(65))

    A^2 = (3 * sqrt(13) + sqrt(65)) * (3 * sqrt(13) - sqrt(65)) * (sqrt(65) + sqrt(13)) * (sqrt(65) - sqrt(13))

    A^2 = (9 * 13 - 65) * (65 - 13)

    A^2 = 13 * (9 - 5) * 13 * (5 - 1)

    A^2 = 13^2 * 4 * 4

    A = 13 * 4

    A = 52

    Same answer, different methods.

  • TomV
    Lv 7
    4 weeks ago

    Simplest approach is that presented by Barkley Hound. But don't forget to subtract the 76 for the 128 to get the shaded area of 52.

    Another, much more tedious approach would be to calculate the dimensions of the triangle.The leftmost leg of the triangle is √(4²+6²) = √(16+36) = √52 = 2√13The rightmost leg of the triangle is √(8² + 12²) = √(64+144) = √208 = 4√13

    The bottom leg of the triangle is √(2² + 16²) = √(4+256) = √260 = 2√65

    Check for right triangle:

     (2√13)² + (4√13)² = 52 + 208 = 260 = 2√65

    We have a right triangle whose base is 4√13 and altitude is 2√13

    Area = bh/2 = (2√13)(4√13)/2 = 4*13 = 52

    And with considerably more effort, we obtain the same answer as Barkley Hound did much more efficiently. 

    Ans: 52

  • 4 weeks ago

    Total area of square is 16x8 = 128

    Now there are 3 right triangles to subtract.

    4 and 6 is 4x6 / 2 = 12

    12 and 8  is 12x8 / 2 = 48

    2 and 16 is 2x16 / 2 = 16

    Total 128 - 76 = 52

    • TomV
      Lv 7
      4 weeks agoReport

      Don't forget to subtract the 76 from the 128 to get the area of 52

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