Problem 1.For any triangle, construct a segment parallel to a base of the triangle that divides the triangle into two equal areas.
HINT: If two similar figures have a ratio K of their areas, what is the ratio of the corresponding sides?
A Geometer's SketchPad file for the construction.
A.If the parallel segment that divides the triangle into two equal areas is drawn for each base, a smaller triangle is formed. What is the ratio of the area of the small triangle to the original?
Hint: Read parts B and C. Identify several similar triangles.
B. What is the ratio of the area of the shaded triangle to the area of the original triangle in the figure below? Here again the segments parallel to the bases divide the original triangles into two equal areas.
C. Prove that the measures of the three shaded areas in each of the figures below are the same. In each figure what is the ratio of the area of one of the regions to the area of the original triangle?
D. Look at the various ratios you have found in parts A, B, and C. Are there any other relationships among areas or combinations of areas?