【Haichuan Teahouse】The annoying little stars
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The side length of square ABCD is 2, and H, G, F, and E are the midpoints of each side respectively. Find the area of the shaded region.
Lazy method: Use CAD to draw it, and the area of the shaded part can be calculated directly
The area of the shadow is the sum of six congruent triangles, equal to 4/3; P
It doesn’t seem to be congruent triangles; the two in the middle are different from those on the sides, but their areas are equal.
The CAD measurement value is 1.3333333……3
Once you understand it, it’s actually just the flexible application of similar triangles. 1. Connecting the several points that are directly opposite each other makes it easier to understand, namely AC, BD, EG, FH ； 2. Find the intersection points O, X, Y ； 3. By using similar triangles, it can be seen that the area of EOY is 1/8 of the total area sought ； 4. By using similar triangles, it can be seen that the areas of EXY and XYO are equal; thus, the area of XYO is 1/16 of the total area sought ； 5. XO = 1/2 * AH; therefore, the height of triangle AHY is 1/2 of the height of triangle XYO ； 6. It is concluded that the area of XYO = 1/2 * base * height / (1+2) * 1/2 = 1/12 ； So, according to item 4, the total area required is 16*1/12=4/3.