A rectangle has length 40 cm and breadth 30 cm, it has an ecternal isosceles triangle with equal sides of 17 cm each along one of its smaller sides as the base of te triangle. Find the pentagon thus formed.

Answer

To find the area of the pentagon formed by a rectangle and an external isosceles triangle, we can follow these steps:

### Step 1: Calculate the area of the rectangle
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]
For the given rectangle:
- Length = 40 cm
- Breadth = 30 cm

So, the area of the rectangle is:
\[ A = 40 \, \text{cm} \times 30 \, \text{cm} = 1200 \, \text{cm}^2 \]

### Step 2: Calculate the semi-perimeter of the isosceles triangle
The semi-perimeter \( s \) of a triangle is given by:
\[ s = \frac{a + b + c}{2} \]
For the isosceles triangle:
- Two equal sides \( a = 17 \, \text{cm} \) and \( b = 17 \, \text{cm} \)
- Base \( c = 30 \, \text{cm} \)

Calculating the semi-perimeter:
\[ s = \frac{17 + 17 + 30}{2} = \frac{64}{2} = 32 \, \text{cm} \]

### Step 3: Calculate the area of the isosceles triangle using Heron's formula
Heron's formula for the area \( A \) of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values:
\[ A = \sqrt{32 \times (32 - 17) \times (32 - 17) \times (32 - 30)} \]
Calculating each term:
- \( s - a = 32 - 17 = 15 \)
- \( s - b = 32 - 17 = 15 \)
- \( s - c = 32 - 30 = 2 \)

Now substituting back:
\[ A = \sqrt{32 \times 15 \times 15 \times 2} \]
Calculating:
\[ A = \sqrt{32 \times 15^2 \times 2} = \sqrt{32 \times 225} = \sqrt{7200} = 120 \, \text{cm}^2 \]

### Step 4: Calculate the area of the pentagon
The area of the pentagon is the sum of the area of the rectangle and the area of the triangle:
\[ \text{Area of Pentagon} = \text{Area of Rectangle} + \text{Area of Triangle} \]
Substituting the values:
\[ \text{Area of Pentagon} = 1200 \, \text{cm}^2 + 120 \, \text{cm}^2 = 1320 \, \text{cm}^2 \]

### Final Answer:
The area of the pentagon is \( 1320 \, \text{cm}^2 \).

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