Question

1. 12, 25, 11
2. 11, 23, 11
3. 11, 4, 6
4. 9, 20, 8
5. 10, 9, 9

state if the three numbers can be the measures of the sides of a triangle.

Explanation:

Step1: Recall triangle inequality theorem

For three side lengths \(a\), \(b\), \(c\) (where \(a\leq b\leq c\)), the sum of the two shorter sides must be greater than the longest side (\(a + b>c\)).

Step2: Check each set

• Set 9) \(12\), \(25\), \(11\)

Let \(a = 11\), \(b=12\), \(c = 25\). Then \(a + b=11 + 12=23\). Since \(23<25\), these cannot be the side lengths of a triangle.

• Set 10) \(11\), \(23\), \(11\)

Let \(a = 11\), \(b = 11\), \(c=23\). Then \(a + b=11+11 = 22\). Since \(22<23\), these cannot be the side lengths of a triangle.

• Set 11) \(11\), \(4\), \(6\)

Let \(a = 4\), \(b = 6\), \(c=11\). Then \(a + b=4 + 6=10\). Since \(10<11\), these cannot be the side lengths of a triangle.

• Set 12) \(9\), \(20\), \(8\)

Let \(a = 8\), \(b = 9\), \(c=20\). Then \(a + b=8 + 9=17\). Since \(17<20\), these cannot be the side lengths of a triangle.

• Set 13) \(10\), \(9\), \(9\)

Let \(a = 9\), \(b = 9\), \(c = 10\). Then \(a + b=9+9 = 18\). Since \(18>10\), these can be the side lengths of a triangle.

Answer:

Only the set \(10\), \(9\), \(9\) (set 13) can be the measures of the sides of a triangle.