Understanding the Triangle Inequality Theorem

To solve this problem, we need to use a fundamental concept in geometry called the Triangle Inequality Theorem. This theorem helps us determine if a set of three lengths can actually form a triangle.

What is the Triangle Inequality Theorem?

The 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. In simpler terms, for a triangle with side lengths $a$, $b$, and $c$, the following three conditions must always be true:

• $a + b > c$
• $a + c > b$
• $b + c > a$

Equivalently, this also implies that the difference between the lengths of any two sides must be less than the length of the third side. For example:

• $|a - b| < c$

Applying the Theorem to the Given Triangle Sides

We are given two sides of a triangle with lengths 5 cm and 1.5 cm. Let these sides be $a = 5$ cm and $b = 1.5$ cm. Let the length of the third side be $c$. We need to find the possible range for $c$ using the Triangle Inequality Theorem.

Using the conditions:

1. Sum of two sides > Third side: 
   The sum of the two given sides is $5 + 1.5 = 6.5$ cm. So, the third side ($c$) must be less than this sum. $$ a + b > c \\ 5 + 1.5 > c \\ 6.5 > c $$ This means $c < 6.5$ cm.
2. Difference of two sides < Third side: 
   The positive difference between the two given sides is $|5 - 1.5| = |3.5| = 3.5$ cm. So, the third side ($c$) must be greater than this difference. $$ |a - b| < c \\ |5 - 1.5| < c \\ 3.5 < c $$ This means $c > 3.5$ cm.

Combining these two conditions ($c < 6.5$ and $c > 3.5$), the length of the third side $c$ must be strictly between 3.5 cm and 6.5 cm.

So, the valid range for the third side is $3.5 \text{ cm} < c < 6.5 \text{ cm}$.

Evaluating the Options for the Third Side

Now let's look at the given options and see which one falls outside this valid range ($3.5 \text{ cm} < c < 6.5 \text{ cm}$).

• Option 1: 3.4 cm 
  Is 3.4 within the range $3.5 < c < 6.5$? No, because 3.4 is not greater than 3.5.
• Option 2: 3.6 cm 
  Is 3.6 within the range $3.5 < c < 6.5$? Yes, because 3.5 < 3.6 < 6.5.
• Option 3: 3.8 cm 
  Is 3.8 within the range $3.5 < c < 6.5$? Yes, because 3.5 < 3.8 < 6.5.
• Option 4: 4.1 cm 
  Is 4.1 within the range $3.5 < c < 6.5$? Yes, because 3.5 < 4.1 < 6.5.

Conclusion

Based on the Triangle Inequality Theorem, the length of the third side must be greater than 3.5 cm. Among the given options, only 3.4 cm is not greater than 3.5 cm. Therefore, a triangle cannot have sides of length 5 cm, 1.5 cm, and 3.4 cm because it violates the condition that the sum of the two shorter sides must be greater than the longest side (1.5 + 3.4 = 4.9, which is not greater than 5). It also violates the difference condition ($|5 - 1.5| = 3.5$, which is not less than 3.4).

Thus, the length of the third side of the triangle cannot be 3.4 cm.