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Area of Outer Triangle composed of two inner triangles

  1. Jul 17, 2017 #1
    1. The problem statement, all variables and given/known data
    1.
    In the figure below, AB=BC=CD. If the area of triangle CDE is 42, what is the area of triangle ADG.

    See the attached figure

    2. Relevant equations
    I think we can start from area of triangle which is given by:

    Area of triangle CDE = ½ * CE * DE

    Or 42 = ½ * CE * DE

    3. The attempt at a solution
    area of outer traingle composed of inner traingles ets.jpg
    I can do some work to transform CE into AD or in my opinion AD = 3 * CE. But the question does not provide any relation between DE & DG. This is what I want to know.


    Somebody please guide me how to solve it.

    Zulfi.
     
  2. jcsd
  3. Jul 17, 2017 #2

    scottdave

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    Hint: They are similar triangles.
     
  4. Jul 17, 2017 #3

    scottdave

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    AD = 3*CD, not CE
     
  5. Jul 17, 2017 #4

    Ray Vickson

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    Use standard properties of similar triangles.
     
  6. Jul 17, 2017 #5

    scottdave

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  7. Jul 19, 2017 #6
    Hi,
    Thanks for your hints. I have done following but i dont think that its close to the solution:
    AB = BC= CD
    or 42 = 1/2 * CE * DE
    84 = CE * DE
    CD^2 = CE^2 + DE^2
    CD/AD = CE/AG = DE/DG

    AD = 3CD
    1/3 = CE/AG = DE/DG

    Some body please guide me.

    Zulfi.
     
  8. Jul 19, 2017 #7

    scottdave

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    You have the information there, if you rearrange it some.

    Take this: AD = 3CD
    and this
    1/3 = DE/DG

    and this: 84 = CE * DE
    What is the formula (in side lengths) for the area of the big triangle? Can you substitute in things (which you can get numerical values for), to solve for a number value of the area of the big triangle?
     
  9. Jul 19, 2017 #8

    Mark44

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    @zak100, here's a different, but one that is related to yours, to help you get some geometric intuition. What's the area of the large rectangle if we are given this information:
    AC = 3 * BC
    CE = 3 * CD
    Area of small rectangle = 10

    Rect.png
     
  10. Jul 19, 2017 #9
    Hi,
    Mr. Mark44. You have provided me a good question. I would try it once i finish my current problem.
    Mr. scottdave: how you got this relationship?
    1/3 = DE/DG

    Can it be applied to bases also?

    Zulfi.
     
  11. Jul 19, 2017 #10

    Mark44

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    This is simple algebra, and comes from the equation DG = 3 * DE, which is equivalent to DE = (1/3) * DG

    Yes, the same idea can be applied.
     
  12. Jul 19, 2017 #11

    Mark44

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    You should look at it now, since you're having so much difficulty with the posted problem. If you understand the relationships between similar triangles and between similar rectangles, the problem I gave can be solved by inspection (no writing needed).
     
  13. Jul 19, 2017 #12

    scottdave

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    You actually had it already, when you wrote: 1/3 = CE/AG = DE/DG
    You should really try the problem which @Mark44 posted with the rectangles (note the drawing is not to scale, though). Once you understand rectangles, solving for triangles is an easy step from that.
     
  14. Jul 20, 2017 #13
    Hi,
    Thanks for your advise. If i leave it, it would be a diversion. I may concatenate from problems to problems. Actually i was not starting with a proper eq. My problem was to find the are of ADG, so i must write its eq. first:

    area of ADG = 1/2 AG * DG
    = 1/2 * 3CE * 3DE
    = 1/2 * 9 * CE * DE
    = 1/2 * 9 * 84 (Note CE * DE = 84 from post 6)
    = 378
    Thanks for your interest and continuous guidance.

    I would now consider the rectangle problem.
    Zulfi.
     
  15. Jul 20, 2017 #14

    scottdave

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    Yes you have the correct answer. Once you complete the rectangle, hopefully you will see a pattern.
     
  16. Jul 23, 2017 #15
    Hi,
    I have solved the rectangle prob:
    Area of large rect = AC * CE
    = 3 * BC * 3 * CD
    = 9 * BC * CD
    Note BC * CD = area of large rect = 10 (given)

    Area of large rectangle = 9 * 10
    = 90

    Thanks for this prob.

    Zulfi.
     
  17. Jul 23, 2017 #16

    Mark44

    Staff: Mentor

    Yes, that's correct. The idea is that since both dimensions of the small rectangle are tripled, the area of the large rectangle will be 3 * 3 = 9 times as large.
    It's exactly the same idea as in your problem with the triangles. That is, the large triangle will have an area 9 times as large as the small upper triangle.
     
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