Question 49

The length of side AB is 6.

In an isosceles triangle with equal sides BC and CA, if the perimeter is 30, then the length of side AB can be determined by setting up the equation based on the perimeter. Given that the two equal sides are represented as x, the equation becomes x + x + AB = 30, allowing us to solve for AB.

A) 4

If side AB were 4, then the equation would be 2x + 4 = 30, leading to 2x = 26, which gives x = 13. However, this value does not satisfy the properties of an isosceles triangle as both equal sides would need to be less than the perimeter divided by 2, making 4 an incorrect choice.

B) 6

Choosing side AB as 6 leads to the equation 2x + 6 = 30. Solving this gives 2x = 24, resulting in x = 12. This configuration is valid for an isosceles triangle, as both equal sides (12) can indeed be greater than the length of AB (6), confirming this as the correct option.

C) 10

If side AB were 10, then the equation would become 2x + 10 = 30, simplifying to 2x = 20, hence x = 10. This scenario creates a degenerate triangle where the two equal sides are equal to the base, which violates the triangle inequality theorem; thus, this option is incorrect.

D) 15

Assuming side AB is 15, we would have the equation 2x + 15 = 30, which gives us 2x = 15, resulting in x = 7.5. This leads to the two equal sides being shorter than the base, again violating the triangle inequality theorem; therefore, this option is not valid.

E) 22

If side AB were 22, we would set up the equation 2x + 22 = 30, giving us 2x = 8, or x = 4. This results in two sides being equal to 4, which is not feasible in a triangle with a base of 22, making this option incorrect.

Conclusion

The correct length of side AB is 6, as it satisfies the triangle's perimeter condition while maintaining the properties of an isosceles triangle. All other options fail due to violations of the triangle inequality theorem or yield configurations that do not align with the given perimeter of 30.