The difference between altitude and base of a right angled triangle is 17 cm and its hypotenuse is 25 cm. What is the sum of the base and altitude of the triangle is :

Options

A. 24 cm
B. 31 cm
C. 34 cm
D. can't be determined

Answer

To solve the problem, we need to find the sum of the base and altitude of a right-angled triangle, given that the difference between the altitude (let's denote it as \( h \)) and the base (let's denote it as \( b \)) is 17 cm, and the hypotenuse (denoted as \( c \)) is 25 cm.

### Step-by-Step Solution:

1. **Set Up the Equations**:
We know two things from the problem:
- The difference between the altitude and the base:
\[ h - b = 17 \quad \text{(1)} \]
- The hypotenuse of the triangle:
\[ c = 25 \quad \text{(2)} \]

2. **Use the Pythagorean Theorem**:
For a right-angled triangle, the Pythagorean theorem states:
\[ h^2 + b^2 = c^2 \quad \text{(3)} \]
Substituting the value of \( c \) from equation (2):
\[ h^2 + b^2 = 25^2 = 625 \quad \text{(4)} \]

3. **Express \( h \) in terms of \( b \)**:
From equation (1), we can express \( h \) as:
\[ h = b + 17 \quad \text{(5)} \]

4. **Substitute \( h \) in Equation (4)**:
Now, substitute equation (5) into equation (4):
\[ (b + 17)^2 + b^2 = 625 \]
Expanding the left side:
\[ b^2 + 34b + 289 + b^2 = 625 \]
Combine like terms:
\[ 2b^2 + 34b + 289 = 625 \]
Rearranging gives:
\[ 2b^2 + 34b - 336 = 0 \quad \text{(6)} \]

5. **Simplify the Quadratic Equation**:
Divide the entire equation by 2:
\[ b^2 + 17b - 168 = 0 \quad \text{(7)} \]

6. **Solve the Quadratic Equation**:
We can use the quadratic formula \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \) where \( A = 1, B = 17, C = -168 \):
\[ b = \frac{-17 \pm \sqrt{17^2 - 4 \cdot 1 \cdot (-168)}}{2 \cdot 1} \]
Calculate the discriminant:
\[ 17^2 = 289 \quad \text{and} \quad -4 \cdot 1 \cdot -168 = 672 \]
Thus,
\[ b = \frac{-17 \pm \sqrt{289 + 672}}{2} \]
\[ b = \frac{-17 \pm \sqrt{961}}{2} \]
\[ b = \frac{-17 \pm 31}{2} \]
This gives us two possible values for \( b \):
\[ b = \frac{14}{2} = 7 \quad \text{and} \quad b = \frac{-48}{2} = -24 \quad \text{(not valid)} \]
Therefore, \( b = 7 \).

7. **Find the Altitude \( h \)**:
Substitute \( b \) back into equation (5):
\[ h = 7 + 17 = 24 \]

8. **Calculate the Sum of Base and Altitude**:
Now, we find the sum:
\[ h + b = 24 + 7 = 31 \text{ cm} \]

### Final Answer:
The sum of the base and altitude of the triangle is **31 cm**.