Thread: 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?

Hello, Sarah!

are midpoints of the sides of .
are the midpoints of the sides of .
The interiors of are shaded.
What fraction of the interior of 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
. . we see that is shaded.

The unshaded triangle, , has the area of
. . and of it is shaded.

That is: .


Therefore, the total shaded area is: .