Question

for each set of three lengths, determine if they can be the side lengths of a triangle.

lengths	can be side lengths of a triangle	cannot be side lengths of a triangle
5, 13, 17	○	○
8, 6, 23	○	○
9, 10, 15	○	○

Explanation:

To determine if three lengths can form a triangle, we use the triangle inequality theorem: the sum of the lengths of any two sides must be greater than the length of the remaining side. We check this for each set of lengths.

Step 1: Analyze \( 15.7, 22.1, 5.6 \)

• Check \( 5.6 + 15.7 \) vs \( 22.1 \):

\( 5.6 + 15.7 = 21.3 \). Since \( 21.3 < 22.1 \), the sum of the two smaller sides is not greater than the largest side.
→ Cannot be side lengths of a triangle.

Step 2: Analyze \( 5, 13, 17 \)

• Check \( 5 + 13 \) vs \( 17 \):

\( 5 + 13 = 18 \). Since \( 18 > 17 \), and by symmetry (checking all pairs: \( 5 + 17 > 13 \), \( 13 + 17 > 5 \)):
→ Can be side lengths of a triangle.

Step 3: Analyze \( 8, 6, 23 \)

• Check \( 6 + 8 \) vs \( 23 \):

\( 6 + 8 = 14 \). Since \( 14 < 23 \), the sum of the two smaller sides is not greater than the largest side.
→ Cannot be side lengths of a triangle.

Step 4: Analyze \( 9, 10, 15 \)

• Check \( 9 + 10 \) vs \( 15 \):

\( 9 + 10 = 19 \). Since \( 19 > 15 \), and by symmetry (checking all pairs: \( 9 + 15 > 10 \), \( 10 + 15 > 9 \)):
→ Can be side lengths of a triangle.

Answer:

Lengths	Can be side lengths of a triangle	Cannot be side lengths of a triangle
\( 5, 13, 17 \)	⚫ (selected)	∘ (unselected)
\( 8, 6, 23 \)	∘ (unselected)	⚫ (selected)
\( 9, 10, 15 \)	⚫ (selected)	∘ (unselected)

(Note: In the table, "⚫" represents the selected option, and "∘" represents the unselected option.)