Question

The perimeter of this isosceles triangle is 22 cm. If one side is 6 cm, what are the possible lengths of the other two sides?

Explain how you know. Provide at least one reason for your answer.

Answer 1

Let the other two sides of the isosceles triangle be x cm each. Since the triangle is isosceles, we know that the two equal sides have the same length.

Therefore, the perimeter of the triangle is:

6 cm + x cm + x cm = 22 cm

Simplifying this equation, we get:

2x + 6 = 22

Subtracting 6 from both sides, we get:

2x = 16

Dividing both sides by 2, we get:

x = 8

So the possible lengths of the other two sides are 8 cm each.

We can also check that this answer makes sense by verifying that the sum of the lengths of any two sides of the triangle is greater than the length of the remaining side. In this case, we have:

6 cm + 8 cm > 8 cm

6 cm + 8 cm > 6 cm

8 cm + 8 cm > 6 cm

All of these inequalities hold true, so we know that the triangle with sides of length 6 cm, 8 cm, and 8 cm is a valid isosceles triangle with a perimeter of 22 cm.

Answer 2

Let's call the length of each of the other two sides of the isosceles triangle "x". Since the triangle is isosceles, the two sides of length "x" are equal to each other.

The perimeter of the triangle is the sum of the lengths of all three sides. So, we can set up an equation:

Perimeter = 22 cm
6 cm + x + x = 22 cm

Simplifying the equation, we get:

2x + 6 = 22
2x = 16
x = 8

So, the length of each of the other two sides could be 8 cm, since the triangle is isosceles and the other two sides are equal to each other.

We can also check that this answer is correct by verifying that the triangle inequality holds. For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

In this case, we have:

6 + 8 > 8
6 + 8 > 6
8 + 8 > 6

All three inequalities hold, so a triangle with sides of length 6 cm, 8 cm, and 8 cm can exist. Therefore, 8 cm is a possible length for each of the other two sides of the isosceles triangle given that one side is 6 cm.