A grade school question from China

My wife showed me a homework problem of a friend’s son in China that he could not solve, and his mother also could not solve. I thought it was quite an interesting problem, so I thought I would share.

I do not have an acceptable diagram (which admittedly would make the problem much easier to digest), so I will be as precise as possible in describing the problem. Suppose that you have a triangle ABC with a right angle at vertex B. The length of the side AC is equal to 14. Construct squares AGFB (read counter-clockwise) on side AB and BEDC on side BC. Draw the line from vertex A to the vertex D of square BCDE, and let H be the intersection of the line segment AD and BC. Suppose further that the line segment GH  is parallel to AC. What is the area of the quadrilateral BHDE?

We let x denote the side length of AB, y denote the side length of BC, so that x^2 + y^2 = 14^2 = 196. Put z for the length of BH. Since AC and GH are parallel, it follows that z = y - x. Therefore the desired area is given by [AED] - [ABH] (the square brackets denote the area of the polygon with the given vertices), or

\frac{y(x+y)}{2} - \frac{x(y-x)}{2} = \frac{x^2 + y^2}{2} = 98.

This problem is nice because it requires adroit use of many different geometric facts, and once the proper principles are applied, the solution is beautiful and elegant. Hard to imagine an 11 year old being able to do this regularly though!

Edit: in the original post there was an error where I forgot to divide by 2 in the penultimate step. Silly mistake!


2 thoughts on “A grade school question from China

  1. leaf.

    You may have missed a factor of 1/2 in the second expression of your final answer. Is this on the standard curriculum or a problem from outside of class? It’s hard to tell…

    1. prayersontests Post author

      Yes I did in fact miss a factor of 2. Thank you for pointing it out!

      And I have no idea if this is standard curriculum or not. The parent of the child couldn’t solve the problem so she asked social media for help, and it eventually got to me.


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