Question 3

The shortest distance between two points is a straight line, which helps us check if a triangle can be formed.

Step 1 — Check the first set of lengths

Let us consider the side lengths 3 cm, 4 cm, and 8 cm. We can imagine these as paths between three points, A, B, and C. Let AB be 4 cm, BC be 3 cm, and AC be 8 cm.

Let us check the path from A to B. The direct path from A to B is side AB's length. $\text{Direct path AB} = \mathbf{4 \text{ cm}}$ The roundabout path from A to B goes through C. This path is the sum of AC and BC. $\text{Roundabout path AC + BC} = \mathbf{8 \text{ cm}} + \mathbf{3 \text{ cm}}$

$\boxed{\mathbf{11 \text{ cm}}}$ We compare the direct path with the roundabout path. Is 4 cm shorter than 11 cm? Yes, it is. The direct path is shorter than the roundabout path. This condition works.

Step 2 — Check the second path

Now, let us check the path from B to C. The direct path from B to C is side BC's length. $\text{Direct path BC} = \mathbf{3 \text{ cm}}$ The roundabout path from B to C goes through A. This path is the sum of BA and AC. $\text{Roundabout path BA + AC} = \mathbf{4 \text{ cm}} + \mathbf{8 \text{ cm}}$

$\boxed{\mathbf{12 \text{ cm}}}$ We compare the direct path with the roundabout path. Is 3 cm shorter than 12 cm? Yes, it is. The direct path is shorter than the roundabout path. This condition also works.

Step 3 — Check the third path

Finally, let us check the path from A to C. The direct path from A to C is side AC's length. $\text{Direct path AC} = \mathbf{8 \text{ cm}}$ The roundabout path from A to C goes through B. This path is the sum of AB and BC. $\text{Roundabout path AB + BC} = \mathbf{4 \text{ cm}} + \mathbf{3 \text{ cm}}$

$\boxed{\mathbf{7 \text{ cm}}}$ We compare the direct path with the roundabout path. Is 8 cm shorter than 7 cm? No, it is not. The direct path is longer than the roundabout path. This means these three points cannot form a triangle.

$\boxed{\text{A triangle cannot exist with sides 3 cm, 4 cm, 8 cm.}}$

Step 4 — Check the second set of lengths

Let us consider the side lengths 2 cm, 3 cm, and 6 cm. Let AB be 3 cm, BC be 2 cm, and AC be 6 cm.

Let us check the path from A to C. The direct path from A to C is side AC's length. $\text{Direct path AC} = \mathbf{6 \text{ cm}}$ The roundabout path from A to C goes through B. This path is the sum of AB and BC. $\text{Roundabout path AB + BC} = \mathbf{3 \text{ cm}} + \mathbf{2 \text{ cm}}$

$\boxed{\mathbf{5 \text{ cm}}}$ We compare the direct path with the roundabout path. Is 6 cm shorter than 5 cm? No, it is not. The direct path is longer than the roundabout path. This means these three points cannot form a triangle.

$\boxed{\text{A triangle cannot exist with sides 2 cm, 3 cm, 6 cm.}}$

Answer

(a) For sides 3 cm, 4 cm, and 8 cm, the direct path of 8 cm is longer than the roundabout path of 4 cm + 3 cm = 7 cm. So, a triangle cannot exist. (b) For sides 2 cm, 3 cm, and 6 cm, the direct path of 6 cm is longer than the roundabout path of 3 cm + 2 cm = 5 cm. So, a triangle cannot exist.