Problem C. 991.

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C. 991. A right-angled triangle of sides 3, 4, 5 is cut into two parts with a line perpendicular to the hypotenuse. One part is a quadrilateral that has an inscribed circle, and the other part is a right-angled triangle. Find the lengths of the sides of the quadrilateral.

(5 points)

Deadline expired.

Sorry, the solution is published in Hungarian only.

Megoldás. Legyen a=3, b=4, c=5.

1.eset: Az egyenes a b oldalt metszi a Q_b pontban.

Legyen BP=x, ekkor PA=5-x. Mivel $AQ_bP\triangle\approx ABC\triangle$ , ezért $\frac{PQ_b}{5-x}=\frac34~\Rightarrow~PQ_b=\frac34(5-x)$ és $\frac{Q_bA}{5-x}=\frac54~\Rightarrow~Q_bA=\frac54(5-x)$ .

Ekkor $CQ_b=4-\frac54(5-x)=\frac{5x-9}{4}$ .

Mivel BCQ_bP érintőnégyszög, azért BC+Q_bP=CQ_b+PB, azaz $3+\frac34(5-x)=\frac{5x-9}{4}+x$ , ebből x=3.

Tehát a négyszög oldalai 3; 1,5; 1,5; 3.

2. eset: Az egyenes az a oldalt metszi a Q_a pontban. Legyen AP=x, ekkor PB=5-x. Mivel $BQ_aP\triangle\approx BAC\triangle$ , ezért $\frac{PQ_a}{5-x}=\frac43~\Rightarrow~PQ_a=\frac43(5-x)$ és $\frac{Q_aB}{5-x}=\frac53~\Rightarrow~Q_aB=\frac53(5-x)$ .

Ekkor $CQ_a=3-\frac53(5-x)=\frac{5x-16}{3}$ .

Mivel APQ_aC érintőnégyszög, azért AC+Q_aP=CQ_a+PA, $4+\frac43(5-x)=\frac{5x-16}{3}+x$ , ebből x=4.

Tehát a négyszög oldalai 4; $\frac43$ ; $\frac43$ ; 4.

Statistics on problem C. 991.

123 students sent a solution.

5 points: 76 students.

4 points: 5 students.

3 points: 30 students.

2 points: 8 students.

1 point: 3 students.

Unfair, not evaluated: 1 solution.

Problems in Mathematics of KöMaL, May 2009

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