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Math Help - Area of a triangle

  1. #1
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    Area of a triangle

    Hello. I am stuck on this problem

    We are given triangle ABC and point E on segment AC, such that <ABE = 75 deg and <EBC= 75 deg. Also, AB = 4 and BC = 6. Find the areas of triangles ABE and EBC.

    Now at first I thought that the angle bisector formed a right angle but now I think it doesn't. By the angle bisector theorem, we can say that \displaystyle \frac{4}{AE}=\frac{6}{EC} but that doesn't get me any further. I can find the area of the triangle by using the law of cosines but that also doesn't help me. I feel like the fact that there is an angle bisector should be a big part of the solution, but I'm stumped.

    Any hints?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by griffen View Post
    Hello. I am stuck on this problem

    We are given triangle ABC and point E on segment AC, such that <ABE = 75 deg and <EBC= 75 deg. Also, AB = 4 and BC = 6. Find the areas of triangles ABE and EBC.

    Now at first I thought that the angle bisector formed a right angle but now I think it doesn't. By the angle bisector theorem, we can say that \displaystyle \frac{4}{AE}=\frac{6}{EC} but that doesn't get me any further. I can find the area of the triangle by using the law of cosines but that also doesn't help me. I feel like the fact that there is an angle bisector should be a big part of the solution, but I'm stumped.

    Any hints?
    area\Delta ABC=\frac{1}{2}*AB*BC*sin C=\frac{1}{2}*4*6*sin 150^o

    \frac{AE}{EC}=\frac{4}{6}\implies\frac{area\Delta ABE}{area\Delta EBC}=\frac{4}{6}

    Can you finish off?
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  3. #3
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    Thanks!

    I got before that the area could be calculated that way, I just didn't know how to use it. I get the area of ABC to be 6. Now if the legs of the triangles are in that ratio and have the same height, then the areas also are in the same ratio. That was the insight I was missing!

    2x+3x=6

    The ratio reduces to 3:2, so 3x+4x=6 for some x since the two smaller triangles add up to the bigger one. This yields x=\frac{6}{5} and this means that the two triangles have areas \frac{12}{5},\frac{18}{5}

    Look good?
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  4. #4
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    That isn't right I think now. I'm stuck again. I don't see how to relate the fact that the area of ABE+EBC=ABC and that AE/EC=4/6. I've been thinking all day. Can I have a hint please? I've searched online for a while too and can't see anything that's clicking.
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by griffen View Post
    That isn't right I think now. I'm stuck again. I don't see how to relate the fact that the area of ABE+EBC=ABC and that AE/EC=4/6. I've been thinking all day. Can I have a hint please? I've searched online for a while too and can't see anything that's clicking.
    Why are you stuck? Your answer in post #3 is correct.
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  6. #6
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    Quote Originally Posted by alexmahone View Post
    Why are you stuck? Your answer in post #3 is correct.
    Oh wow, haha. I guess I just don't see how \displaystyle \frac{AE}{EC}=\frac{4}{6}\implies\frac{area\Delta ABE}{area\Delta EBC}=\frac{4}{6}
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by griffen View Post
    Oh wow, haha. I guess I just don't see how \displaystyle \frac{AE}{EC}=\frac{4}{6}\implies\frac{area\Delta ABE}{area\Delta EBC}=\frac{4}{6}
    Draw a diagram. Note that triangles ABE and EBC have the same altitude BX (say).

    \displaystyle \frac{area\Delta ABE}{area\Delta EBC}=\frac{\frac{1}{2}*AE*BX}{\frac{1}{2}*EC*{BX}}  =\frac{AE}{EC}
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