Question

chang knows one side of a triangle is 13 cm. which set of two sides is possible for the lengths of the other two sides of this triangle?
○ 5 cm and 8 cm
○ 6 cm and 7 cm
○ 7 cm and 2 cm
○ 8 cm and 9 cm

Explanation:

Step1: Recall Triangle Inequality Theorem

For a triangle with sides \(a\), \(b\), \(c\), the sum of any two sides must be greater than the third side. Let the given side be \(c = 13\) cm, and the other two sides be \(a\) and \(b\). We need to check \(a + b>c\), \(a + c>b\), and \(b + c>a\). Since \(c = 13\) is the largest side (we can assume for checking, or check all cases), the key inequality here is \(a + b>13\) (because \(a + 13>b\) and \(b + 13>a\) will always hold if \(a\) and \(b\) are positive, which they are as side lengths).

Step2: Check each option

• Option 1: \(5 + 8 = 13\). But \(5 + 8\) is not greater than \(13\) (it's equal), so this does not satisfy the triangle inequality.
• Option 2: \(6+7 = 13\). Again, the sum is equal to \(13\), not greater, so this does not satisfy.
• Option 3: \(7 + 2=9\). \(9<13\), so this does not satisfy.
• Option 4: \(8 + 9 = 17\). \(17>13\), so this satisfies the triangle inequality (and the other inequalities \(8 + 13>9\) and \(9 + 13>8\) also hold as \(21>9\) and \(22>8\)).

Answer:

8 cm and 9 cm