【Haichuan Teahouse】Brain-teasing Series: It is said to be possible to calculate it by hand directly
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This post was last edited by Desert Fish on 2021-8-2 07:46. It is an exam question from an elite high school in Qingdao; top students were able to solve it mentally, while average students struggled greatly and were unable to complete it. As shown, the area of square ABCD is 36, and PQ is the trisection point; find the area of the shaded region in the figure.
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Triangles PQR and ARD are similar, with the ratio of their corresponding heights being 1:3. Since the area of the square is 36, its side length is 6; therefore, PQ = 2. The height of the smaller triangle is 1.5, so the area of the shaded region is 1.5
The similar triangles ARD and PQD are similar; PQ is 1/3 of the side length. Therefore, the height of the shaded area is 1/4 of the side length, which equals 6*1/4=1.5. Since PQ equals 6/3=2, the area is 2*1.5/2=1.5.