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anonymous

  • one year ago

Find the area & perimeter of the figure

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  1. anonymous
    • one year ago
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  2. mathstudent55
    • one year ago
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    You were given a triangle like this: |dw:1437006808180:dw|

  3. mathstudent55
    • one year ago
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    What do the circled marks mean? |dw:1437006848302:dw|

  4. anonymous
    • one year ago
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    @11.seventeen what do the circled marks mean

  5. anonymous
    • one year ago
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    doesn't it mean its equal

  6. anonymous
    • one year ago
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    @mathstudent55

  7. mathstudent55
    • one year ago
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    Yes. All sides are congruent. That means every side of this triangle measures 2a mm. That makes the perimeter easy to find. What is the perimeter of a triangle?

  8. mathstudent55
    • one year ago
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    \(\Large P_{triangle} = a + b + c\) where a, b, and c are the lengths of the sides of the triangle.

  9. mathstudent55
    • one year ago
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    \(P = 2a + 2a + 2a = 6a\) The perimeter of the triangle is 6a mm

  10. mathstudent55
    • one year ago
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    Ok so far?

  11. anonymous
    • one year ago
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    Yes. It's the same as what i got @mathstudent55

  12. mathstudent55
    • one year ago
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    Now you have to find the area.

  13. anonymous
    • one year ago
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    And I would do that how ? Because I don't have the heigh . I have the base is 2

  14. mathstudent55
    • one year ago
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    To find the area, you need to find the height of the triangle since the formula for the area involves the height.

  15. mathstudent55
    • one year ago
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    Have you heard about the ratios of the lengths of the sides of a 30-60-90 triangle?

  16. anonymous
    • one year ago
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    Yes but I never understood it

  17. mathstudent55
    • one year ago
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    Ok. I'll explain it to you. Here is a triangle with a right angle and a 30-deg angle and a 60-deg angle. |dw:1437011007751:dw|

  18. mathstudent55
    • one year ago
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    Since the right angle is 90 deg, this triangles' three angles have measures 30, 60, and 90 degrees. This is what is called a 30-60-90 triangle.

  19. mathstudent55
    • one year ago
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    In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the short leg. If you call the length of the short leg 1, then the hypotenuse has length 2. |dw:1437011223749:dw|

  20. mathstudent55
    • one year ago
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    The length of the long leg is \(\sqrt 3\) times the length of the short leg.

  21. mathstudent55
    • one year ago
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    |dw:1437011291789:dw|

  22. mathstudent55
    • one year ago
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    So far all we have is that the lengths of the sides of a 30-60-90 triangle are in the ratio of 1 : \(\sqrt 3 : 2\)

  23. mathstudent55
    • one year ago
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    Do you understand so far?

  24. anonymous
    • one year ago
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    sort of

  25. mathstudent55
    • one year ago
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    It is simpler than you think. Think of \(\sqrt 3\) as being approximately 1.7 All this means is that: In a 30-60-90 triangle, the long leg is 1.7 times the length of the short leg. The hypotenuse is 2 times the length of the short leg.

  26. mathstudent55
    • one year ago
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    If a 30-60-90 triangle has a short leg of 1, then the long leg is 1.7 * 1 = 1.7 the hypotenuse is 2 * 1 = 2

  27. anonymous
    • one year ago
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    how do I get the height out of this?

  28. mathstudent55
    • one year ago
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    Another example: If a 30-60-90 triangle has a short leg of length 4, then the long leg is 1.7 * 4 = 6.8 and the hypotenuse is 2 * 4 = 8

  29. mathstudent55
    • one year ago
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    Ok. Let's get back to our problem.

  30. mathstudent55
    • one year ago
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    |dw:1437011694076:dw|

  31. mathstudent55
    • one year ago
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    You see our triangle above. Since all sides are congruent, all angles are also congruent. That means all angles measure 60 degrees. Ok?

  32. mathstudent55
    • one year ago
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    |dw:1437011786387:dw|

  33. mathstudent55
    • one year ago
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    We are looking for the length of the height. By drawing the height, we created two new triangles. Notice each small triangle is a 30-60-90 triangle.

  34. mathstudent55
    • one year ago
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    |dw:1437011853170:dw|

  35. mathstudent55
    • one year ago
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    Look at the figure above. Each of the two small triangles is a 30-60-90 triangle. The 30 degree angle is above. Since the side of the large triangle has length 2a, half of that is just a, and that is the length of the short side of the 30-60-90 triangle.

  36. mathstudent55
    • one year ago
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    Look at the small triangle in the circle and ignore the rest. |dw:1437012005959:dw|

  37. mathstudent55
    • one year ago
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    The short leg has length a. The long leg has length \(\sqrt 3\) times a, which is simply: \(\sqrt 3 a\) |dw:1437012059638:dw|

  38. mathstudent55
    • one year ago
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    |dw:1437012157126:dw|

  39. mathstudent55
    • one year ago
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    The long leg of the small triangle is the height of the large triangle. That was the height we needed for the area.

  40. mathstudent55
    • one year ago
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    |dw:1437012211126:dw|

  41. anonymous
    • one year ago
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    so is the height 3a?

  42. mathstudent55
    • one year ago
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    Now we have all the info we need to find the area of the large triangle.

  43. mathstudent55
    • one year ago
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    The height is \(\sqrt 3 a\), not 3a. \(\Large A_{triangle} = \dfrac{bh}{2} \)

  44. mathstudent55
    • one year ago
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    \(\Large A = \dfrac{(a)(\sqrt 3 a)}{2} = \dfrac{\sqrt 3}{2}a^2 ~mm^2\)

  45. anonymous
    • one year ago
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    thanks!!!

  46. mathstudent55
    • one year ago
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    You're welcome.

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