\( 3 \mathrm{~cm}, 4 \mathrm{~cm} \) and 7 cm are not the bengths of the triangle. Give the reasen.

To determine if three lengths can form a triangle, we use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, if we check: 1. \( 3 \text{ cm} + 4 \text{ cm} = 7 \text{ cm} \), which is not greater than \( 7 \text{ cm} \). 2. \( 3 \text{ cm} + 7 \text{ cm} = 10 \text{ cm} \), which is greater than \( 4 \text{ cm} \). 3. \( 4 \text{ cm} + 7 \text{ cm} = 11 \text{ cm} \), which is greater than \( 3 \text{ cm} \). Since the first inequality does not hold, these lengths cannot form a triangle. Now, let's think about it like this: imagine trying to create a triangle puzzle with sticks of these lengths. If you take the 3 cm and 4 cm sticks, they can barely touch the tip of your 7 cm stick, creating a straight line instead of closing up a triangle. The sides aren't long enough to connect and "close the loop," hence no triangle can be formed!