Question

can the three side lengths make a triangle?

7, 2, 6

12, 6, 8

8, 11, 22

8, 2, 6

11, 15, 11

Explanation:

Apply the Triangle Inequality Theorem to each set of side lengths

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side. Equivalently, the sum of the two shorter sides must be greater than the longest side: \(a + b > c\), where \(c\) is the longest side.

Evaluate each given set of side lengths

• For \(7, 2, 6\): Shorter sides are \(2\) and \(6\).

$$2 + 6 = 8 > 7 \quad (\text{Yes})$$

• For \(12, 6, 8\): Shorter sides are \(6\) and \(8\).

$$6 + 8 = 14 > 12 \quad (\text{Yes})$$

• For \(8, 11, 22\): Shorter sides are \(8\) and \(11\).

$$8 + 11 = 19 gtr 22 \quad (\text{No})$$

• For \(8, 2, 6\): Shorter sides are \(2\) and \(6\).

$$2 + 6 = 8 gtr 8 \quad (\text{No})$$

• For \(11, 15, 11\): Shorter sides are \(11\) and \(11\).

$$11 + 11 = 22 > 15 \quad (\text{Yes})$$

Answer:

Side Lengths	Can they make a triangle?
\(12, 6, 8\)	Yes
\(8, 11, 22\)	No
\(8, 2, 6\)	No
\(11, 15, 11\)	Yes