【Haichuan Teahouse】Brain-teasing Series: Said to be a turning point
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Question: The side length of square ABCD is 12, and the radius of the incircle of triangle EDC is 3. Find the area of the shaded region.
How was it calculated? Was it done using CAD drawing methods again? ? Middle school students don’t know CAD
Calculating the area of the triangle: By connecting the incenter O to the 3 vertices, the two legs have lengths 3+9 and 3+X, while the hypotenuse has length 9+x (using the method of the inscribed circle and the angle-bisector theorem). Applying the Pythagorean theorem yields x=6; the area of the triangle is therefore 54, and 144–54=90
Mainly, I can’t recall the properties of the incircle of a triangle, so I couldn’t derive that the hypotenuse equals 9+x. If it is known that the angle between a side and the center of the circle is a right angle, then using the property of congruent triangles, it can be determined that the hypotenuse equals 9+x; thus, X=6:victory: