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Difficulty: HardTriangle Properties and Angle Theorems

An isosceles triangle has two sides of length 55 and 1111. A second triangle has side lengths of 1212, 1818, and dd, where dd is an integer. If dd is equal to the perimeter of the first triangle, what is the perimeter of the second triangle?

  1. A
    41
  2. B
    46
  3. C
    51
  4. 57Answer
  5. E
    63

Answer

57
To find the perimeter of the second triangle, we must first determine the value of dd, which is the perimeter of the first triangle. The first triangle is isosceles with two sides of length 55 and 1111. By the Triangle Inequality Theorem, the sum of any two side lengths must exceed the third. A triangle with sides 5,5,115, 5, 11 is impossible because 5+5=10<115 + 5 = 10 < 11. Thus, the sides of the first triangle must be 11,11,511, 11, 5, giving a perimeter of 11+11+5=2711 + 11 + 5 = 27. This means d=27d = 27. The second triangle has sides of length 1212, 1818, and 2727. Since 12+18=30>2712 + 18 = 30 > 27, this is a valid triangle. Its perimeter is 12+18+27=5712 + 18 + 27 = 57.

Step-by-Step Solution

1
Determine the possible side lengths of the first isosceles triangle.
The sides must be 1111, 1111, and 55.
An isosceles triangle has two equal sides. The side lengths must be either 5,5,115, 5, 11 or 11,11,511, 11, 5. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. If the sides were 5,5,115, 5, 11, then 5+5=10<115 + 5 = 10 < 11, which violates this theorem. Thus, the only valid side lengths are 1111, 1111, and 55 (since 5+11=16>115 + 11 = 16 > 11).
2
Calculate the perimeter of the first triangle to find the value of dd.
d=27d = 27
The perimeter of the first triangle is the sum of its three sides: 11+11+5=2711 + 11 + 5 = 27. Since dd is equal to this perimeter, d=27d = 27.
3
Verify that a triangle with side lengths 1212, 1818, and 2727 is valid.
The triangle is valid.
We check the Triangle Inequality Theorem: 12+18=30>2712 + 18 = 30 > 27, 12+27=39>1812 + 27 = 39 > 18, and 18+27=45>1218 + 27 = 45 > 12. Since all inequalities hold, the second triangle is valid.
4
Calculate the perimeter of the second triangle.
Perimeter = 5757
The perimeter of the second triangle is the sum of its side lengths: 12+18+27=5712 + 18 + 27 = 57.

Key Concept

Triangle Inequality Theorem and Isosceles Triangle Properties
Estimated Time:2m 0s
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Topics

Triangle Properties and Angle Theorems

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