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
7, 25, 18 | ○ | ○
5, 18, 21 | ○ | ○
4, 10, 10 | ○ | ○
23.6, 9.5, 15.0 | ○ | ○

Explanation:

Step1: Apply triangle inequality rule

The triangle inequality theorem states that for three lengths $a, b, c$ (sorted so $c$ is largest), $a + b > c$ must hold. If $a + b = c$, they form a degenerate triangle (not a valid non-degenerate triangle).

Step2: Test 7,25,18

Sort: $7, 18, 25$. Calculate sum: $7 + 18 = 25$. Since $25 = 25$, it fails the inequality.

Step3: Test 5,18,21

Sort: $5, 18, 21$. Calculate sum: $5 + 18 = 23$. Since $23 > 21$, it satisfies the inequality.

Step4: Test 4,10,10

Sort: $4, 10, 10$. Calculate sum: $4 + 10 = 14$. Since $14 > 10$, it satisfies the inequality.

Step5: Test 23.6,9.5,15.0

Sort: $9.5, 15.0, 23.6$. Calculate sum: $9.5 + 15.0 = 24.5$. Since $24.5 > 23.6$, it satisfies the inequality.

Answer:

Lengths	Can be side lengths of a triangle	Cannot be side lengths of a triangle
5, 18, 21	$\boldsymbol{\bigcirc}$
4, 10, 10	$\boldsymbol{\bigcirc}$
23.6, 9.5, 15.0	$\boldsymbol{\bigcirc}$