Triangle Inequality Theorem Homework online exercise for | Live ... - Free Printable

Final Answer:
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

──────────────────────────────────────

Explanation:
- To check if three side lengths can form a triangle, you need to test all three combinations.
- For example, with sides 13, 9, and 4:
→ 13 + 9 = 22 > 4 ✔️
→ 13 + 4 = 17 > 9 ✔️
→ 9 + 4 = 13 — this is NOT greater than 13 (it’s equal), so ✘ it does NOT form a triangle.
- For sides 24, 17, and 30:
→ 24 + 17 = 41 > 30 ✔️
→ 24 + 30 = 54 > 17 ✔️
→ 17 + 30 = 47 > 24 ✔️ → ✔ It DOES form a triangle.
- For sides 8, 8, 8:
→ All sums are 16 > 8 → ✔ It forms a triangle (equilateral).

For the “circle the possible third side” problems:
- Use the rule: (larger side - smaller side) < third side < (larger side + smaller side)
- Example: sides 6 cm and 10 cm → third side must be between 4 cm and 16 cm → so 15 is OK, 16 is not (must be less than 16), 17 is too big → only 15 is correct.

For “range of possible lengths”:
- Subtract the two known sides → that’s the smallest possible value (but not including it).
- Add the two known sides → that’s the largest possible value (but not including it).
- Example: sides 5 and 8 → range is greater than 3 and less than 13 → so 3 < x < 13.

For “is X in the range?”:
- Find the range first, then see if the number fits inside.
- Example: sides 20 and 24 → range is 4 < x < 44 → 4 is NOT included → so 4 is NOT in the range.

For “how many possible lengths?”:
- The third side must be a whole number between the lower and upper bounds.
- Example: sides 18 and 20 → range is 2 < x < 38 → so possible whole numbers are 3, 4, 5, ..., 37 → that’s 35 numbers.

──────────────────────────────────────

Teach the Topic:
Triangle inequality is like a rule for building triangles. Imagine you have three sticks. You can only make a triangle if no single stick is as long or longer than the other two sticks put together. If one stick is too long, the other two can’t reach each other to close the shape.

Simple example:
Can you make a triangle with sticks that are 3 inches, 4 inches, and 8 inches?
→ 3 + 4 = 7 — that’s less than 8 → NO, you can’t make a triangle.
But with 3, 4, and 5?
→ 3 + 4 = 7 > 5 → YES! That works.

──────────────────────────────────────

Practice Task:
Can the side lengths 7, 5, and 10 form a triangle? Show your work using inequality statements.