Question

what is the largest perimeter of an isosceles triangle with side lengths of 30 and 40?
110
largest: 4, 9 *
your answer
smallest: 9, 4 *
your answer
what is the smallest perimeter of an isosceles triangle with side lengths of 30 and 40?
your answer

Explanation:

Step1: Consider two cases for the isosceles triangle

Case 1: The equal - sides (legs) are \(30\). Then the sides of the triangle are \(30\), \(30\), \(40\).
Case 2: The equal - sides (legs) are \(40\). Then the sides of the triangle are \(40\), \(40\), \(30\).

Step2: Check the triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), \(c\), the triangle inequality theorem states that \(a + b>c\), \(a + c>b\), and \(b + c>a\).

• In Case 1: \(30+30 = 60>40\), \(30 + 40=70>30\), \(30+40 = 70>30\)
• In Case 2: \(40 + 40=80>30\), \(40+30 = 70>40\), \(40 + 30=70>40\)

Step3: Calculate the perimeters

• Perimeter of Case 1: \(P_1=30 + 30+40=100\)
• Perimeter of Case 2: \(P_2=40 + 40+30=110\)

Answer:

The largest perimeter is \(110\) and the smallest perimeter is \(100\)