【Haichuan Teahouse】Brain-teasing Series: Teachers say they’re easy questions, but students claim they lack the necessary conditions
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As shown in the figure, in a quarter-circle, a circle is tangent to CD and OB at points F and E respectively. It is given that the length of CD is 12, and CD is parallel to OB. Find the area of the shaded region. Friends, do you think conditions are lacking? What are some good solutions? This was a final exam question from a renowned experimental middle school in Shanghai. Due to its high difficulty, less than 1% of the students in the entire grade were able to solve it completely; most students gave up as they felt they lacked the necessary information to do so. The teacher originally thought it was an easy question; such exam results almost made the teacher so angry that he could have vomited blood
Complete the circle (with diameter R, the length of the hypotenuse according to the Pythagorean theorem, and the radius of the small circle). Let the diameter of the small circle be r; then 4 times the area of the shadow (with the full circle completed) equals PI(12*12 + 2r*2r) – 4*PI*r*r, which is equal to 144PI. The area of the shadow is therefore 36PI
The condition of just CD=12 is not sufficient; imagine what happens if the quarter circle is a bit larger or smaller……
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For this type of question, there is only one way to approach it: since the area is not variable, expressing it in algebraic form is the correct solution.
Well, this must be the correct answer; I assumed wrongly that the area could change.