The altitude of a right-angled triangle is 7 cm less than its base . If the hypotenuse is 13 cm , then find the other two sides .

Answer

To solve the problem, we will denote the base of the right-angled triangle as \( b \) cm and the altitude (height) as \( h \) cm. According to the problem, the altitude is 7 cm less than the base, which can be expressed as:

\[ h = b - 7 \]

We also know that the hypotenuse \( c \) is 13 cm. By the Pythagorean theorem, we have:

\[ c^2 = b^2 + h^2 \]

Substituting the value of \( c \) and \( h \) into the equation, we get:

\[ 13^2 = b^2 + (b - 7)^2 \]

Now, let's simplify this equation step by step.

1. Calculate \( 13^2 \):
\[ 169 = b^2 + (b - 7)^2 \]

2. Expand \( (b - 7)^2 \):
\[ (b - 7)^2 = b^2 - 14b + 49 \]

3. Substitute this back into the equation:
\[ 169 = b^2 + b^2 - 14b + 49 \]

4. Combine like terms:
\[ 169 = 2b^2 - 14b + 49 \]

5. Rearrange the equation to set it to zero:
\[ 2b^2 - 14b + 49 - 169 = 0 \]
\[ 2b^2 - 14b - 120 = 0 \]

6. Divide the entire equation by 2 to simplify:
\[ b^2 - 7b - 60 = 0 \]

7. Now, we will factor the quadratic equation:
\[ (b - 12)(b + 5) = 0 \]

8. Set each factor to zero:
\[ b - 12 = 0 \quad \text{or} \quad b + 5 = 0 \]

This gives us:
\[ b = 12 \quad \text{or} \quad b = -5 \]

Since the base cannot be negative, we take \( b = 12 \) cm.

9. Now, substitute \( b \) back to find \( h \):
\[ h = b - 7 = 12 - 7 = 5 \text{ cm} \]

Thus, the two sides of the triangle are:
- Base \( b = 12 \) cm
- Altitude \( h = 5 \) cm

### Final Answer:
The other two sides of the triangle are 12 cm (base) and 5 cm (height).