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K. 195. The isosceles triangle ABC has a right angle at A; the length of its legs is AB=AC=4 cm. The circles with centers at B and C of radius 4 cm intersect the hypotenuse at D and E, respectively. What is the area of the shaded domain? (See the Figure.)

(6 points)

This problem is for grade 9 students only.

Deadline expired on 10 February 2009.


Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás. A satírozott területet jelölje T. Ezen kívül keletkezik még két egybevágó alakzat, ezek területe felírható: t=t_{\triangle}-\frac{45}{360}\cdot4^2\pi=\frac{4\cdot4}{2}-2\pi=8-2\pi.

Így a terület:

T=t_{\triangle}-2t=8-2(8-2\pi)=4\pi-8\approx 4,5664.


Statistics on problem K. 195.
170 students sent a solution.
6 points:55 students.
5 points:50 students.
4 points:30 students.
3 points:12 students.
2 points:7 students.
1 point:1 student.
0 point:8 students.
Unfair, not evaluated:7 solutions.


  • Problems in Mathematics of KöMaL, January 2009

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