The perimeter of triangle ABC is 68 cm. Find the lengths of the sides of this triangle if AB: BC = 2: 3 and BC: AC = 6: 7.

According to the condition of the problem, the length of the side AB of the triangle ABC relates to the length of the side BC as 2: 3, therefore:

| AB | / | BC | = 2/3,

or

| AB | = (2/3) * | BC |.

It is also known that the length of the side BC of the triangle ABC refers to the length of the side AC as 6: 7, therefore:

| BC | / | AC | = 6/7,

or

| AC | = (7/6) * | BC |.

By the condition of the problem, the perimeter of the triangle ABC is 68 cm, therefore:

| AB | + | BC | + | AC | = 68.

Substituting into this ratio the values ​​| AB | = (2/3) * | BC | and | AC | = (7/6) * | BC |, we get:

(2/3) * | BC | + | BC | + (7/6) * | BC | = 68.

We solve the resulting equation:

| BC | * (2/3 + 1 + 7/6) = 68;

| BC | * (4/6 + 6/6 + 7/6) = 68;

| BC | * (17/6) = 68;

| BC | = 68 / (17/6);

| BC | = 68 * (6/17);

| BC | = 6 * 68/17;

| BC | = 6 * 4;

| BC | = 24 cm.

Knowing the length of the side BC, we find | AB | and | AC |:

| AB | = (2/3) * | BC | = (2/3) * 24 = 16 cm;

| AC | = (7/6) * | BC | = (7/6) * 24 = 28 cm.

Answer: | AB | = 16 cm; | BC | = 24 cm; | AS | = 28 cm.