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Math Help - Triangle Ratio Problem

  1. #1
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    Triangle Ratio Problem

    Given a triangle PQR, A, B, and C are midpoints of the sides of triangle PQR, and M, N, and O are the midpoints of the sides of triangle ABC. The interiors of triangle PAB, triangle AQC, triangle BCR, and triangle MNO are shaded. What fraction of the interior of triangle PQR is shaded?

    I don't know how/why the answer would be 13/16?
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  2. #2
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    Hello, Sarah!

    A, B,\text{ and }C are midpoints of the sides of \Delta PQR.
    M, N,\text{ and }O are the midpoints of the sides of \Delta ABC.
    The interiors of \Delta PAB,\:\Delta AQC,\:\Delta BCR,\text{ and }\Delta MNO are shaded.
    What fraction of the interior of \Delta PQR is shaded?
    Code:
                      P
                      *
                     /:\
                    /:::\
                   /:::::\
                  /:::::::\
                 /::::O::::\
               A* - - * - - *B
               /:\   /:\   /:\
              /:::\ /:::\ /:::\
             /::::M*- - -*N::::\
            /:::::::\   /:::::::\ 
           /:::::::::\-/:::::::::\
          * - - - - - * - - - - - *
          Q           C           R
    It can be shown that:
    . . When the midpoints of the sides of a triangle are joined,
    . . the triangle is divided into four congruent triangles.
    Each triangle has one-fourth of the area.

    Joining the midpoints of the side of \Delta PQR
    . . we see that \frac{3}{4}\text{ of }\Delta PQR is shaded.

    The unshaded triangle, \Delta ABC, has \frac{1}{4} the area of \Delta PQR.
    . . and \frac{1}{4} of it is shaded.

    That is: . \Delta MNO \:=\:\left(\frac{1}{4}\right)\left(\frac{1}{4}\rig  ht)\:=\:\frac{1}{16}\, \text{ of }\Delta PQR


    Therefore, the total shaded area is: . \frac{3}{4} + \frac{1}{16} \:=\:\boxed{\frac{13}{16}}\,\text{ of }\Delta PQR.

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