Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 298. (September 2011)

K. 298. Three consecutive sides of a quadrilateral have the same length. The two interior angles enclosed by the three sides are 60o and 70o. What is the measure of the largest angleThree consecutive sides of a quadrilateral have the same length. The two interior angles enclosed by the three sides are 60o and 70o. What is the measure of the largest angle of the quadrilateral? of the quadrilateral?

(6 pont)

Deadline expired on October 10, 2011.

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

Megoldás. Használjuk az ábra jelöléseit.

A $\displaystyle BCD$ háromszög szabályos, ezért $\displaystyle BD=a$ és ez az átló a $\displaystyle 70^\circ$-os szöget $\displaystyle 10^\circ$ és $\displaystyle 60^\circ$-os szögekre bontja. Ezek szerint $\displaystyle ABD$ háromszög egyenlőszárú, és alapján fekvő szögei $\displaystyle 85^\circ$-osak. Ezért a négyszög legnagyobb szöge $\displaystyle 85^\circ + 60^\circ = 145^\circ$.

Statistics:

 282 students sent a solution. 6 points: 98 students. 5 points: 87 students. 4 points: 31 students. 3 points: 9 students. 2 points: 5 students. 1 point: 2 students. 0 point: 47 students. Unfair, not evaluated: 3 solutions.

Problems in Mathematics of KöMaL, September 2011