One side of a triangle is 5 cm and the other side is 10 cm and its perimeter is P cm, where P is an integer. The least and the greatest possible values of P are respectively

Question

Accepted Answer

To determine the least and greatest possible values of the perimeter $ P $ of a triangle with sides 5 cm and 10 cm, we need to consider the properties of triangles, specifically the triangle inequality theorem. This 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.

Let's denote the third side of the triangle as $ x $ cm. The perimeter $ P $ of the triangle is given by:

$$ P = 5 + 10 + x = 15 + x $$

### Least Possible Value of $ P $

1. **Triangle Inequality for the Third Side:**
   - For the side $ x $, the inequality $ 5 + x > 10 $ must hold true. Simplifying this gives:
     $$ x > 5 $$

2. **Minimum Integer Value for $ x $:**
   - Since $ x $ must be greater than 5, the smallest integer value for $ x $ is 6.

3. **Calculating the Minimum Perimeter:**
   - Substituting $ x = 6 $ into the perimeter equation:
     $$ P = 15 + 6 = 21 $$

### Greatest Possible Value of $ P $

1. **Triangle Inequality for the Third Side:**
   - For the side $ x $, the inequality $ 10 + x > 5 $ is always true for positive $ x $.
   - The inequality $ 5 + 10 > x $ simplifies to:
     $$ x < 15 $$

2. **Maximum Integer Value for $ x $:**
   - Since $ x $ must be less than 15, the largest integer value for $ x $ is 14.

3. **Calculating the Maximum Perimeter:**
   - Substituting $ x = 14 $ into the perimeter equation:
     $$ P = 15 + 14 = 29 $$

### Conclusion

The least possible value of $ P $ is 21, and the greatest possible value of $ P $ is 29. Therefore, the correct answer is Option 3: 21 and 29.

### Explanation of Incorrect Options

- **Option 1 (19 and 29):** The minimum perimeter cannot be 19 because the smallest integer value for $ x $ that satisfies the triangle inequality is 6, leading to a minimum perimeter of 21.
- **Option 2 (20 and 28):** The minimum perimeter cannot be 20 for the same reason as above, and the maximum perimeter cannot be 28 because the largest integer value for $ x $ is 14, leading to a maximum perimeter of 29.
- **Option 4 (22 and 27):** The minimum perimeter cannot be 22 because the smallest integer value for $ x $ is 6, leading to a minimum perimeter of 21. Similarly, the maximum perimeter cannot be 27 because the largest integer value for $ x $ is 14, leading to a maximum perimeter of 29.

Thus, Option 3 is the only correct choice, as it accurately reflects the possible range of the perimeter based on the given conditions.