In right angle `Delta ABC`, BD divides `Delta ABC ` into two equal perimeter `Delta `. Find AD if AC = 10 cm , BC = 8 cm ` angle B = 90^(@)`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Triangle
We are given a right triangle \( \Delta ABC \) where:
- \( AC = 10 \, \text{cm} \)
- \( BC = 8 \, \text{cm} \)
- \( \angle B = 90^\circ \)

### Step 2: Find the Length of Side \( AB \)
Using the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ AB^2 + 8^2 = 10^2 \]
\[ AB^2 + 64 = 100 \]
\[ AB^2 = 100 - 64 = 36 \]
Taking the square root:
\[ AB = \sqrt{36} = 6 \, \text{cm} \]

### Step 3: Set Up the Perimeter Equation
Let \( AD = x \) and \( DB = y \). Since \( BD \) divides \( \Delta ABC \) into two triangles with equal perimeters, we can write the perimeter of both triangles:
- Perimeter of \( \Delta ABD = AB + AD + BD = 6 + x + y \)
- Perimeter of \( \Delta CBD = BC + CD + BD = 8 + (10 - x) + y \)

### Step 4: Set the Perimeters Equal
Set the two perimeters equal:
\[ 6 + x + y = 8 + (10 - x) + y \]
Simplifying this equation:
\[ 6 + x + y = 18 - x + y \]
Subtracting \( y \) from both sides:
\[ 6 + x = 18 - x \]
Adding \( x \) to both sides:
\[ 6 + 2x = 18 \]
Subtracting 6 from both sides:
\[ 2x = 12 \]
Dividing by 2:
\[ x = 6 \, \text{cm} \]

### Step 5: Conclusion
Since \( AD = x \), we find that:
\[ AD = 6 \, \text{cm} \]

Thus, the final answer is:
\[ \text{AD} = 6 \, \text{cm} \]

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