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The problem describes a right-angled triangle where:

- The hypotenuse ("कर्ण") is 13 cm.
- The perpendicular height ("शीर्षलम्ब") is 7 cm less than the base ("आधार").

We need to find the values of the base and height and match them to one of the provided options.

### Step-by-step solution:

1. **Let the base be** \( x \) cm.
2. **The height will be** \( (x - 7) \) cm because it is 7 cm less than the base.

3. According to the Pythagoras theorem:
   \[
   \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2
   \]
   Substituting the given values:
   \[
   13^2 = x^2 + (x - 7)^2
   \]
   Simplifying:
   \[
   169 = x^2 + (x - 7)^2
   \]
   Expanding \( (x - 7)^2 \):
   \[
   169 = x^2 + (x^2 - 14x + 49)
   \]
   \[
   169 = 2x^2 - 14x + 49
   \]
   Subtract 49 from both sides:
   \[
   120 = 2x^2 - 14x
   \]
   Dividing the entire equation by 2:
   \[
   60 = x^2 - 7x
   \]
   Rearranging the terms:
   \[
   x^2 - 7x - 60 = 0
   \]
   Now, solve this quadratic equation using the quadratic formula:
   \[
   x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-60)}}{2(1)}
   \]
   Simplifying:
   \[
   x = \frac{7 \pm \sqrt{49 + 240}}{2}
   \]
   \[
   x = \frac{7 \pm \sqrt{289}}{2}
   \]
   \[
   x = \frac{7 \pm 17}{2}
   \]
   So, \( x = \frac{7 + 17}{2} = 12 \) or \( x = \frac{7 - 17}{2} = -5 \).

Since base cannot be negative, \( x = 12 \) cm.

4. **Height is** \( x - 7 = 12 - 7 = 5 \) cm.

Thus, the base is 12 cm and the height is 5 cm.

### Correct answer: 
**Option B: आधार = 12 cm, शीर्षलम्ब = 5 cm**

Would you like more details or have any other questions?

### Related Questions:
1. How can we modify the equation for different triangle conditions?
2. What happens if the hypotenuse changes to a different value?
3. Can the quadratic formula always solve such geometry problems?
4. How would the calculation differ if both the base and height were reduced by the same amount?
5. How can this be solved using factorization instead of the quadratic formula?

### Tip:
In problems involving Pythagoras theorem, always check if the values can be part of a known Pythagorean triplet (like 5, 12, 13) to simplify calculations.

एक समकोण त्रिभुज का शीर्षलम्ब इसके आधार से 7 cm कम है यदि कर्ण 13 cm हो तो अन्य दो भुजाएँ है–

Solve a right-angled triangle problem where the hypotenuse is 13 cm, and the height is 7 cm less than the base. Learn how to find the base and height using the Pythagoras theorem and quadratic formula. Suitable for students in Grades 8-10.

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