Two triangles have all their angles equal; of their sides, two pairs are equal, but the third sides are different. Calculate the different sides if the lengths of the equal sides are 80 and 100 units, respectively.
If the angles of two triangles are equal, then the two triangles are similar: the ratio of the corresponding sides is equal.

The lengths of the sides of one triangle are: $80,100, x$; in the other, the sides are $80,100, y$.

The equal sides cannot be corresponding to each other, because then $x=y$ would be true; however, according to our assumption, $x \neq y$.

Accordingly, the 80 unit side of the first triangle corresponds to the 100 or y unit side of the other. Let's take the first case; then the corresponding sides of the two triangles are in the order of

$$
80,100, x \text { and } 100, y, 80
$$

Already now

$$
80: 100=100: y=x: 80
$$

From this

$$
y=125, \quad x=64
$$

Accordingly, the sides of the two triangles, arranged by size:

$64,80,100$ and $80,100,125$.

The sides form a geometric progression, whose ratio is: $\frac{5}{4}$.

We arrive at the same two triangles if the order of the corresponding sides is:

$$
80,100, x \quad \text { and } \quad y, 80,100
$$

that is

$$
80: y=100: 50=x: 100
$$

Now $y=64$ and $x=125$ and the sides of the two triangles are:

$80,100,125$, and $64,80,100$.

Csuri Vilmos (Kossuth Lajos g. VII. o. Pestszenterzsébet.)[^0]


[^0]:    ${ }^{1}$ If the 80-unit side of the first triangle corresponds to the 100-unit side of the second, the 100-unit side of the first triangle cannot correspond to the 80-unit side of the second!