An isosceles right angled triangle has an area 18cm^2 . The semi perimeter of triangle is:class 9 correct ans
12+6root 2cm
6+3 root 2 cm
3+6 root 2 cm
6+6root 2 cm

Let's solve the problem step by step.

### Given:
- The triangle is an **isosceles right-angled triangle**.
- The area of the triangle is \(18 \, \text{cm}^2\).
  
### To Find:
- The semi-perimeter of the triangle.

### Steps:

1. **Identify the Sides:**
   - In an isosceles right-angled triangle, the two legs are equal in length. Let each leg be \(a\).
   - The hypotenuse \(h\) can be found using the Pythagorean theorem:
     \[
     h = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}
     \]

2. **Area of the Triangle:**
   - The area of the triangle is given by:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
     \]
   - Since the triangle is isosceles and right-angled, both base and height are \(a\). Therefore:
     \[
     18 = \frac{1}{2} \times a \times a
     \]
     \[
     18 = \frac{1}{2} a^2
     \]
     \[
     a^2 = 36
     \]
     \[
     a = \sqrt{36} = 6 \, \text{cm}
     \]

3. **Calculate the Hypotenuse:**
   \[
   h = 6\sqrt{2} \, \text{cm}
   \]

4. **Calculate the Perimeter:**
   - The perimeter \(P\) of the triangle is the sum of all its sides:
     \[
     P = a + a + h = 6 + 6 + 6\sqrt{2} = 12 + 6\sqrt{2} \, \text{cm}
     \]

5. **Calculate the Semi-Perimeter:**
   - The semi-perimeter \(s\) is half of the perimeter:
     \[
     s = \frac{P}{2} = \frac{12 + 6\sqrt{2}}{2} = 6 + 3\sqrt{2} \, \text{cm}
     \]

### Conclusion:
The correct semi-perimeter of the triangle is \(6 + 3\sqrt{2}\) cm. The correct answer is **6 + 3√2 cm**.

Would you like any further details or have any questions? Here are some related questions you might find helpful:

1. How do you calculate the area of different types of triangles?
2. What are the properties of isosceles triangles?
3. How is the Pythagorean theorem applied in various geometric problems?
4. What is the significance of the semi-perimeter in triangle calculations?
5. How do different triangle types affect perimeter and area calculations?

**Tip:** For isosceles right-angled triangles, the hypotenuse is always the leg length multiplied by \( \sqrt{2} \), making calculations easier.

Learn how to calculate the semi-perimeter of an isosceles right-angled triangle with an area of 18 cm². Understand the application of the Pythagorean theorem and the formula for triangle area. Suitable for Grade 9 students.

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