Problem 502. Triangle, Two Squares, Midpoint, Perpendicular, Half the measure. Level: High School, College, SAT Prep.

The drawing shows a triangle ABC with the squares ABDE and BCGH. If M is the midpoint of DH, prove that (1) BM is perpendicular to AC and (2) BM is half the measure of AC.
 
 

 
 
 

Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

George Pólya's 1945 book "How to Solve It, A new aspect of Mathematical Method", is a book describing methods of problem solving. It suggests the following steps when solving a mathematical problem: (1) First, you have to understand the problem. (2) After understanding, then make a plan. (3) Carry out the plan. (4) Look back on your work. How could it be better? 
 

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