Question

9 if two sides of a triangle have lengths of 2 and 7, the length of the third side could be 1) 9 2) 8 3) 5 4) 4 10 if two sides of a triangle have lengths of 2 and 8, the length of the third side could be 1) 10 2) 7 3) 6 4) 4 13 which set of numbers could be the lengths of the sides of an isosceles triangle? 1) {1,1,2} 2) {3,3,5} 3) {3,4,5} 4) {4,4,9} 14 which set of integers could represent the lengths of the sides of an isosceles triangle? 1) {1,1,3} 2) {2,2,5} 3) {3,3,6} 4) {4,4,7}

Explanation:

Step1: Triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\). For an isosceles triangle, at least two sides are equal.

Problem 9:

Let the two given sides be \(a = 2\) and \(b = 7\).

• Lower - bound: \(|a - b|=|2 - 7| = 5\)
• Upper - bound: \(a + b=2 + 7=9\). So \(5

Problem 10:

Let the two given sides be \(a = 2\) and \(b = 8\).

• Lower - bound: \(|a - b|=|2 - 8| = 6\)
• Upper - bound: \(a + b=2+8 = 10\). So \(6

Problem 13:

• Option (1) \(\{1,1,2\}\): \(1 + 1=2\), does not satisfy \(a + b>c\) (triangle inequality).
• Option (2) \(\{3,3,5\}\): \(3+3>5\), \(3 + 5>3\), \(3+5>3\) (isosceles as two sides are \(3\)).
• Option (3) \(\{3,4,5\}\): Not isosceles.
• Option (4) \(\{4,4,9\}\): \(4 + 4<9\), does not satisfy triangle inequality.

Answer:

1. (2) \(8\)
2. (2) \(7\)
3. (2) \(\{3,3,5\}\)