In a triangle ABC , AP is perpendicular to BC . If BC = 112 cm , cot B `=4/3` and cot C `=12/5` , calculate the length of AP

Answer

To find the length of AP in triangle ABC, we will follow these steps:

### Step 1: Understand the given information
We have a triangle ABC where AP is perpendicular to BC. We know:
- BC = 112 cm
- cot B = 4/3
- cot C = 12/5

### Step 2: Relate cotangent to the sides
From trigonometry, we know that:
- cot B = BP / AP
- cot C = CP / AP

### Step 3: Express BP and CP in terms of AP
Using the given values of cot B and cot C:
1. From cot B = 4/3, we can write:
\[ BP = \frac{4}{3} AP \]

2. From cot C = 12/5, we can write:
\[ CP = \frac{12}{5} AP \]

### Step 4: Set up an equation for BC
Since BP + CP = BC, we can substitute the expressions we found:
\[ BP + CP = \frac{4}{3} AP + \frac{12}{5} AP = 112 \]

### Step 5: Find a common denominator and combine the fractions
The least common multiple of 3 and 5 is 15. We will convert both fractions:
\[ \frac{4}{3} AP = \frac{20}{15} AP \]
\[ \frac{12}{5} AP = \frac{36}{15} AP \]
Now we can add them:
\[ \frac{20}{15} AP + \frac{36}{15} AP = \frac{56}{15} AP \]
So we have:
\[ \frac{56}{15} AP = 112 \]

### Step 6: Solve for AP
To isolate AP, we multiply both sides by 15:
\[ 56 AP = 112 \times 15 \]
Now, divide both sides by 56:
\[ AP = \frac{112 \times 15}{56} \]

### Step 7: Simplify the expression
Calculating the right side:
\[ AP = \frac{1680}{56} = 30 \]

### Conclusion
The length of AP is **30 cm**.

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