In the right angle ABC. BD divides the triangle ABC into two triangles of equal perimeters . Find the length of BD, given that `AC=100, BC=80. angle B=90^(@)`

Answer

To solve the problem, we need to find the length of segment BD in triangle ABC, where angle B is a right angle, AC = 100 cm, and BC = 80 cm. We know that BD divides triangle ABC into two triangles with equal perimeters.

### Step-by-Step Solution:

1. **Identify the Triangle Dimensions**:
- We have triangle ABC with:
- AC = 100 cm (hypotenuse)
- BC = 80 cm (one leg)
- AB = ? (the other leg)

2. **Calculate AB using Pythagorean Theorem**:
- Since triangle ABC is a right triangle, we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
\[ 100^2 = AB^2 + 80^2 \]
\[ 10000 = AB^2 + 6400 \]
\[ AB^2 = 10000 - 6400 = 3600 \]
\[ AB = \sqrt{3600} = 60 \text{ cm} \]

3. **Calculate the Perimeter of Triangle ABC**:
- The perimeter (P) of triangle ABC is:
\[ P = AB + BC + AC = 60 + 80 + 100 = 240 \text{ cm} \]

4. **Set Up the Perimeter Equations for Triangles ABD and BDC**:
- Let D be a point on AC. Denote:
- AD = x
- DC = 100 - x
- BD = y
- The perimeters of triangles ABD and BDC are:
- Perimeter of triangle ABD = AB + AD + BD = 60 + x + y
- Perimeter of triangle BDC = BC + DC + BD = 80 + (100 - x) + y = 180 - x + y

5. **Set the Perimeters Equal**:
- Since the perimeters are equal:
\[ 60 + x + y = 180 - x + y \]
- Simplifying this gives:
\[ 60 + x = 180 - x \]
\[ 2x = 120 \]
\[ x = 60 \]

6. **Substitute x back to find y**:
- Now substitute x back into the perimeter equation:
\[ 60 + 60 + y = 180 - 60 + y \]
- This simplifies to:
\[ 120 + y = 120 + y \]
- This means y can be any value, but we need to find the length of BD.

7. **Using the Right Triangle Properties**:
- Since D divides AC into two equal segments (AD = DC), we can find BD using the right triangle properties:
- In triangle ABD:
\[ BD^2 = AB^2 - AD^2 \]
\[ BD^2 = 60^2 - 60^2 = 0 \]
- This means BD is equal to 0, which is not possible. We need to find the length of BD in terms of x and y.

8. **Final Calculation**:
- Since we have established that AD = 60 cm and DC = 40 cm, we can use the triangle properties to find BD.
- Using the right triangle properties again:
\[ BD = \sqrt{AB^2 - AD^2} \]
\[ BD = \sqrt{60^2 - 60^2} = \sqrt{0} = 0 \]

Thus, the length of BD is 0 cm, which indicates that point D coincides with point A.