ABC is a `Delta ` BD = 8 BC = 20 CD = 16 ` angle CBD = angle CAB ` find perimeter of ABC

Answer

To find the perimeter of triangle ABC given the conditions in the problem, we can follow these steps:

### Step 1: Understand the Triangle Configuration
We have triangle ABC with point D on side AC. The lengths are given as:
- BD = 8
- BC = 20
- CD = 16
- Angle CBD = Angle CAB

### Step 2: Set Up the Relationship Using Congruence
Since angle CBD is equal to angle CAB, triangles ABD and BDC are similar by the Angle-Angle (AA) criterion. This means:
\[ \frac{AB}{BC} = \frac{BD}{CD} \]

### Step 3: Substitute Known Values
Now, substituting the known values into the proportion:
- AB = ?
- BC = 20
- BD = 8
- CD = 16

The equation becomes:
\[ \frac{AB}{20} = \frac{8}{16} \]

### Step 4: Simplify the Ratio
Simplifying the right side:
\[ \frac{8}{16} = \frac{1}{2} \]
So the equation now looks like:
\[ \frac{AB}{20} = \frac{1}{2} \]

### Step 5: Solve for AB
Cross-multiplying gives:
\[ AB = 20 \times \frac{1}{2} = 10 \]

### Step 6: Find AC
Next, we can find AC using the relationship in triangle BDC:
Since BD + CD = AC,
\[ AC = BD + CD = 8 + 16 = 24 \]

### Step 7: Calculate the Perimeter
Now we can find the perimeter of triangle ABC:
\[ \text{Perimeter} = AB + BC + AC \]
Substituting the known values:
\[ \text{Perimeter} = 10 + 20 + 24 = 54 \]

### Final Answer
The perimeter of triangle ABC is **54**.

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