Two sides of a right triangle measure 15 cm and 17 cm . Which of the following statements can be true of the length of the third side of the triangle ?
I. It is between 4 cm and 7 cm .
II. It is between 20 cm and 23 cm .
III. It is less than 10 cm

Options

A. Only II
B. I and III
C. I and II
D. II and III

Answer

To determine the possible lengths of the third side of a right triangle when two sides measure 15 cm and 17 cm, we can use the properties of right triangles and the triangle inequality theorem.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
We have two sides of a right triangle, which we will denote as:
- Side 1 (L) = 15 cm
- Side 2 (B) = 17 cm

2. **Use the Pythagorean theorem**:
In a right triangle, the relationship between the lengths of the sides is given by:
\[ H^2 = L^2 + B^2 \]
where H is the hypotenuse. Since 17 cm is greater than 15 cm, we can assume that 17 cm is the hypotenuse.

3. **Calculate the hypotenuse**:
\[ H^2 = 15^2 + 17^2 = 225 + 289 = 514 \]
\[ H = \sqrt{514} \approx 22.7 \text{ cm} \]

4. **Determine the range for the third side**:
The third side (let's denote it as C) must satisfy the triangle inequality theorem:
- The sum of the lengths of any two sides must be greater than the length of the third side.
- Therefore, we have:
- \( C + 15 > 17 \) → \( C > 2 \)
- \( C + 17 > 15 \) → \( C > -2 \) (always true)
- \( 15 + 17 > C \) → \( C < 32 \)

Combining these inequalities, we find:
\[ 2 < C < 32 \]

5. **Evaluate the statements**:
- **Statement I**: It is between 4 cm and 7 cm. (True, since 4 < C < 7 falls within 2 < C < 32)
- **Statement II**: It is between 20 cm and 23 cm. (True, since 20 < C < 23 falls within 2 < C < 32)
- **Statement III**: It is less than 10 cm. (True, since C can be less than 10 cm as long as it is greater than 2 cm)

### Conclusion:
All three statements can be true regarding the length of the third side of the triangle.