【Haichuan Teahouse】Brain-teasing Series: Placement of the podiums
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As shown in the figure, there are 3 squares with a side length of 6, arranged as depicted. Find the area of the shaded region
So fast. Let’s start with an extreme case: move the square at the top to the right end. In this way, the area of the triangle can be obtained
When the top square moves from one end to the other, is it a fixed value or a variable value?
This post was last edited by Yang Qing on 2021-8-2 at 13:17 – Two Pairs of Congruent Triangles
It’s similar triangles again; just move the one in the upper right corner. Since the angles and base lengths are equal, the areas must also be equal. 6*(6+6)/2=36.
Then why go to the trouble? Just form a triangle – there’s no need to make a square!
The approach is similar: to form a triangle, it’s sufficient to prove one pair of triangles are congruent, while for a square, two pairs of triangles need to be shown to be congruent. It’s still quite obvious in essence, and not too troublesome. It might be difficult to describe this in words: first, using right angles, alternate interior angles, and side lengths, it is shown that the two blue triangles are congruent; similarly, it is shown that the two yellow triangles are also congruent. It does seem that proving it for just one province is the simpler approach.
Yes, just drag down the two symmetric triangles. . . . . .