Question 5

For three lengths to form a triangle, the sum of any two sides must be greater than the third side.

Step 1 — Check lengths 2, 2, 5

Let us identify the lengths. The lengths are 2, 2, and 5. This rule is called the Triangle Inequality Theorem. We can simplify this check. We only need to see if the sum of the two shortest lengths is greater than the longest length. If this is true, the other two conditions will also be true automatically. The two shortest lengths are 2 and 2. The longest length is 5. Now, we add the two shortest lengths.

$2 + 2$

$= 4$

We compare this sum with the longest length. We check if 4 is greater than 5.

$4 > 5$

This statement is false. So, these lengths cannot form a triangle.

$\boxed{\text{2, 2, 5 cannot be sidelengths of a triangle.}}$

Step 2 — Check lengths 3, 4, 6

Let us identify the lengths. The lengths are 3, 4, and 6. The two shortest lengths are 3 and 4. The longest length is 6. Now, we add the two shortest lengths.

$3 + 4$

$= 7$

We compare this sum with the longest length. We check if 7 is greater than 6.

$7 > 6$

This statement is true. So, these lengths can form a triangle.

$\boxed{\text{3, 4, 6 can be sidelengths of a triangle.}}$

Step 3 — Check lengths 2, 4, 8

Let us identify the lengths. The lengths are 2, 4, and 8. The two shortest lengths are 2 and 4. The longest length is 8. Now, we add the two shortest lengths.

$2 + 4$

$= 6$

We compare this sum with the longest length. We check if 6 is greater than 8.

$6 > 8$

This statement is false. So, these lengths cannot form a triangle.

$\boxed{\text{2, 4, 8 cannot be sidelengths of a triangle.}}$

Step 4 — Check lengths 5, 5, 8

Let us identify the lengths. The lengths are 5, 5, and 8. The two shortest lengths are 5 and 5. The longest length is 8. Now, we add the two shortest lengths.

$5 + 5$

$= 10$

We compare this sum with the longest length. We check if 10 is greater than 8.

$10 > 8$

This statement is true. So, these lengths can form a triangle.

$\boxed{\text{5, 5, 8 can be sidelengths of a triangle.}}$

Step 5 — Check lengths 10, 20, 25

Let us identify the lengths. The lengths are 10, 20, and 25. The two shortest lengths are 10 and 20. The longest length is 25. Now, we add the two shortest lengths.

$10 + 20$

$= 30$

We compare this sum with the longest length. We check if 30 is greater than 25.

$30 > 25$

This statement is true. So, these lengths can form a triangle.

$\boxed{\text{10, 20, 25 can be sidelengths of a triangle.}}$

Step 6 — Check lengths 10, 20, 35

Let us identify the lengths. The lengths are 10, 20, and 35. The two shortest lengths are 10 and 20. The longest length is 35. Now, we add the two shortest lengths.

$10 + 20$

$= 30$

We compare this sum with the longest length. We check if 30 is greater than 35.

$30 > 35$

This statement is false. So, these lengths cannot form a triangle.

$\boxed{\text{10, 20, 35 cannot be sidelengths of a triangle.}}$

Step 7 — Check lengths 24, 26, 28

Let us identify the lengths. The lengths are 24, 26, and 28. The two shortest lengths are 24 and 26. The longest length is 28. Now, we add the two shortest lengths.

$24 + 26$

$= 50$

We compare this sum with the longest length. We check if 50 is greater than 28.

$50 > 28$

This statement is true. So, these lengths can form a triangle.

$\boxed{\text{24, 26, 28 can be sidelengths of a triangle.}}$

Answer

(a) 2, 2, 5 cannot be the sidelengths of a triangle. (b) 3, 4, 6 can be the sidelengths of a triangle. (c) 2, 4, 8 cannot be the sidelengths of a triangle. (d) 5, 5, 8 can be the sidelengths of a triangle. (e) 10, 20, 25 can be the sidelengths of a triangle. (f) 10, 20, 35 cannot be the sidelengths of a triangle. (g) 24, 26, 28 can be the sidelengths of a triangle.