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

Let’s write down a short statement of the problem:

P = 68 cm,

AB: BC = 2: 3, BC: AC = 6: 7.

AB =?

BC =?

AC =?

Solution.

Let us express BC and AC through AB.

According to the main property of the proportion, we can write that 3AB = 2BC and 7BC = 6AC.

Then BC = 3AB / 2,

AC = 7BC / 6 = (7 * 3AB) / (6 * 2) = 7AB / 4.

The perimeter of a triangle (P) is:

P = AB + BC + AC = AB + 3AB / 2 + 7AB / 4 = (4AB + 6AB + 7AB) / 4 = 17AB / 4.

17AB / 4 = 68,

17AB = 272,

AB = 272: 17,

AB = 16.

Hence, AB = 16 cm.

Then BC = 3 * 16/2 = 24 cm, AC = 7 * 16/4 = 28 cm.

Answer: AB = 16 cm, BC = 24 cm, AC = 28 cm.