In the isosceles triangle above, the lengths of sides BC and CA are equal. If the perimeter of the triangle is 30, what is the length of side AB?

Rationale

The length of side AB is 6.

In an isosceles triangle, two sides are equal in length, and the perimeter is the sum of all three sides. If the perimeter is 30 and the two equal sides are denoted as BC and CA, we can determine the length of side AB by setting up the equation: 2x + y = 30, where x is the length of the equal sides and y is the length of side AB.

A (4) YOUR ANSWER

If AB were 4, then the equation 2x + 4 = 30 would imply that 2x = 26, leading to x = 13. This means the lengths of sides BC and CA would be 13, which is not equal to AB, violating the property of an isosceles triangle.

B (6) CORRECT

Setting AB to 6 gives us the equation 2x + 6 = 30, which simplifies to 2x = 24, resulting in x = 12. This means sides BC and CA would each be 12, satisfying the condition of the isosceles triangle where the two sides are equal.

C (10) YOUR ANSWER

If AB were 10, then the equation 2x + 10 = 30 leads to 2x = 20, resulting in x = 10. Although this would make BC and CA equal to AB, it contradicts the prior assumption that AB must be distinct from the equal sides in an isosceles triangle, as AB would not be uniquely defined.

D (15) YOUR ANSWER

With AB set to 15, the equation 2x + 15 = 30 simplifies to 2x = 15, giving x = 7.5. However, this value for x results in non-integer side lengths, which is not typically feasible for the sides of a triangle.

E (22) YOUR ANSWER

If AB were 22, the equation 2x + 22 = 30 would imply that 2x = 8, leading to x = 4. This means sides BC and CA would be 4, which is much smaller than AB, again violating the isosceles triangle property where the base is usually the shortest side.

Conclusion

In the isosceles triangle, the equal sides (BC and CA) must be correctly determined based on the given perimeter. The only viable solution that maintains the properties of an isosceles triangle while satisfying the perimeter constraint is when side AB measures 6, allowing the lengths of the two equal sides to each be 12.

In the isosceles triangle above, the lengths of sides BC and CA are equal. If the perimeter of the triangle is 30, what is the length of side AB?

The length of side AB is 6.

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