Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 268. (November 2010)

K. 268. How long is the radius of the circle in which a chord of length 6 units is twice as far away from the centre as a chord of length 12 units?

(6 pont)

Deadline expired on December 10, 2010.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A húrok szimmetriatengelye és a végpontokba a kör középpontjából húzott szakaszok derékszögű háromszögeket határoznak meg (ld. ábra), melyekben Pithagorasz tételét felírva: \(\displaystyle 4d^2 + 9 =r^2 = d^2 +36\), ahonnan \(\displaystyle d^2=9\), azaz (\(\displaystyle d>0\) miatt) \(\displaystyle d=3\). Ezért a kör sugara \(\displaystyle \sqrt 5\), a két húr távolsága pedig \(\displaystyle 3\sqrt 5\).


Statistics:

204 students sent a solution.
6 points:77 students.
5 points:45 students.
4 points:13 students.
3 points:32 students.
2 points:15 students.
1 point:1 student.
0 point:12 students.
Unfair, not evaluated:9 solutions.

Problems in Mathematics of KöMaL, November 2010