1. The perimeter of a triangle is $210 \mathrm{~cm}$ and the lengths of its sides are in the ratio $3: 5: 7$. Calculate the lengths of the sides of this triangle. Is this triangle a right triangle?
1. Given that the sides of a triangle are in the ratio $a: b: c=3: 5: 7$, there exists a (real) number $k$ such that $a=3k, b=5k$, and $c=7k$.

Then $3k + 5k + 7k = 210$, from which $k=14$.

1 POINT

The sides of the triangle are $42 \text{ cm}, 70 \text{ cm}$, and $98 \text{ cm}$.

3 POINTS

A triangle is right-angled if the lengths of its sides satisfy the Pythagorean theorem.

Since $42^2 + 70^2 = 6664$, and $98^2 = 9604$, the triangle is not right-angled.

2 POINTS

TOTAL 6 POINTS

Note: The student can show a more general statement, i.e., that no triangle with side lengths in the ratio 3:5:7 is right-angled. Indeed, $9k^2 + 25k^2 = 34k^2 \neq 49k^2, k>0$.