Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem C. 1094. (October 2011)

C. 1094. In an isosceles right-angled triangle ABC, H is the point closer to C that divides the leg BC in a 3:1 ratio. G is the point on leg CA such that CG:GA=3:2. Find the measure of the angle enclosed by the line segments HA and GB.

(5 pont)

Deadline expired on November 10, 2011.

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

Megoldás. Legyen $\displaystyle GBC\sphericalangle = \beta$ és $\displaystyle AHC\sphericalangle = \gamma$, a keresett szög pedig $\displaystyle \varphi=\gamma - \beta$. A szokásos módon jelöljük az oldalakat. Ekkor $\displaystyle \tan \beta = \frac{\frac 35 b}{a}=\frac 35$, $\displaystyle \tan \gamma = \frac{b}{\frac 14 a}=4$ szerint

$\displaystyle \tan\varphi=\tan(\gamma - \beta)=\frac{4-\frac 35}{1+4\cdot \frac 35}=1.$

A $\displaystyle HA$ és $\displaystyle GB$ által bezárt szög $\displaystyle 45^\circ$.

### Statistics:

 314 students sent a solution. 5 points: 59 students. 4 points: 183 students. 3 points: 22 students. 2 points: 21 students. 1 point: 5 students. 0 point: 19 students. Unfair, not evaluated: 5 solutions.

Problems in Mathematics of KöMaL, October 2011