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Math Help - Ratios of areas

  1. #1
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    Ratios of areas

    Brat is a trapezoid with M the midpoint of one of the legs. Show that the are of trianlge CAT is equal to the area of BRAT. Picture attached
    Attached Thumbnails Attached Thumbnails Ratios of areas-capture.png  
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  2. #2
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    I have no idea how to go about this problem so any advice would be greatly appreciated.
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  3. #3
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    If you show that the two smaller triangles, MRA and MBC, have the same area, then you're done, I think.

    What can you say about the measures of angles RMA and BMC?

    What can you say about the measures of angles MCB and MAR?

    What then can you say about the third angles of each of the two smaller triangles?

    What then can you say about these two triangles?

    By definition of "midpoint", what can you say about the lengths of segments (that is, of sides) MR and MB?

    What then can you conclude about the two smaller triangles?

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  4. #4
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    Well angles RMA and CMB are congruent and segments MR and MB are congruent, but that still doesnt proove that the two triangles are congruent.
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  5. #5
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    Now look at the other angles mentioned, and consider what this tells you about the two triangles. (It's not "congruency" yet -- you get that from the midpoint consideration -- but it's close enough that the midpoint will complete the process.)
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  6. #6
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    So with Angles RMA and CMB are congruent, segments MR and MB are congruent and segments MC and RA are congruent which makes the triangles congruent so with subtraction of triangle MRA the area of triangle cat is equal to the area of BRAT?
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  7. #7
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    Quote Originally Posted by lamas9000 View Post
    ...and segments MC and RA are congruent...
    How did you arrive at this conclusion?

    Instead, try following the step-by-step instructions provided earlier:

    Look at the second pair of angles listed. What can you conclude, using the fact that the top and bottom sides of the trapezoid are (what sort of) lines?

    Once you have two pairs of congruent angles, what can you say about the third pair of angles?

    What does this say about the two triangles?

    Once you have those sorts of triangles, what does the congruency of one included side say?

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