Find the perimeter of the adjoining figure.
Given : AB = 12 cm, AC = 13 cm . DE = FG = 5 cm , EF = 10 cm and GD = 4 cm .
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Answer

To find the perimeter of the given figure, we will follow these steps:

### Step 1: Identify the sides of the triangle ABC
We are given:
- \( AB = 12 \, \text{cm} \)
- \( AC = 13 \, \text{cm} \)

We need to find the length of side \( BC \).

### Step 2: Apply the Pythagorean theorem
Since triangle ABC is a right triangle, we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + BC^2 \]
Calculating the squares:
\[ 169 = 144 + BC^2 \]
Now, isolate \( BC^2 \):
\[ BC^2 = 169 - 144 \]
\[ BC^2 = 25 \]
Taking the square root:
\[ BC = 5 \, \text{cm} \]

### Step 3: Identify the sides of the quadrilateral ABCH
The quadrilateral ABCH is a rectangle, where:
- Length \( AB = 12 \, \text{cm} \)
- Breadth \( BC = 5 \, \text{cm} \)

### Step 4: Calculate the perimeter of rectangle ABCH
The perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (\text{Length} + \text{Breadth}) \]
Substituting the values:
\[ P_{ABCH} = 2 \times (12 + 5) = 2 \times 17 = 34 \, \text{cm} \]

### Step 5: Add the lengths of the other sides
We are also given:
- \( DE = 5 \, \text{cm} \)
- \( FG = 5 \, \text{cm} \)
- \( EF = 10 \, \text{cm} \)
- \( GD = 4 \, \text{cm} \)

Now, we can calculate the total perimeter of the entire figure:
\[ \text{Total Perimeter} = P_{ABCH} + DE + FG + EF + GD \]
Substituting the values:
\[ \text{Total Perimeter} = 34 + 5 + 5 + 10 + 4 \]
Calculating:
\[ \text{Total Perimeter} = 34 + 24 = 58 \, \text{cm} \]

### Final Answer
The perimeter of the figure is \( 58 \, \text{cm} \).

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