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 for each option if \(a + b>13\), \(a + 13>b\), and \(b + 13>a\) (the last two are always true if \(a + b>13\) since \(13\) is positive and \(a,b\) are positive lengths). So we just check \(a + b>13\).

Step2: Check Option 1 (5 cm and 8 cm)

Calculate the sum: \(5 + 8=13\). But the sum must be greater than \(13\) (not equal), so this is not valid.

Step3: Check Option 2 (6 cm and 7 cm)

Calculate the sum: \(6 + 7 = 13\). Again, sum is equal to \(13\), not greater, so invalid.

Step4: Check Option 3 (7 cm and 2 cm)

Calculate the sum: \(7+2 = 9\). \(9<13\), so invalid.

Step5: Check Option 4 (8 cm and 9 cm)

Calculate the sum: \(8 + 9=17\). \(17>13\), so this satisfies the triangle inequality.

Answer:

8 cm and 9 cm