Let side a,b and c of `DeltaABC` be related by the relation a : b : c = 3 : 5 : 4. Altitudes AD,BE and CF are dropped on BC, CA and AB, respectively. If `P_(1)D+P_(2)E+P_(3)F=42`, then the value of a + b + c is

Options

A. 1200
B. 120
C. 12
D. none of these

Explanation

To solve the problem, we need to find the value of \( a + b + c \) given the ratio of the sides of triangle \( \Delta ABC \) and the equation involving the altitudes. Let's break down the solution step by step.

### Step 1: Define the sides of the triangle
Given the ratio \( a : b : c = 3 : 5 : 4 \), we can express the sides in terms of a variable \( k \):
- \( a = 3k \)
- \( b = 5k \)
- \( c = 4k \)

### Step 2: Identify the midpoints and altitudes
Let \( P_1, P_2, P_3 \) be the midpoints of sides \( BC, CA, AB \) respectively. The altitudes from vertices \( A, B, C \) to the opposite sides are \( AD, BE, CF \).

### Step 3: Write the equation involving the altitudes
We are given:
\[ P_1D + P_2E + P_3F = 42 \]
We need to express \( P_1D, P_2E, P_3F \) in terms of \( a, b, c \).

### Step 4: Calculate \( P_1D \)
Using the properties of triangles and the midpoints:
\[ P_1D = \frac{c - b}{2} = \frac{4k - 5k}{2} = \frac{-k}{2} = \frac{5k - 4k}{2} = \frac{k}{2} \]

### Step 5: Calculate \( P_2E \)
Similarly,
\[ P_2E = \frac{a - c}{2} = \frac{3k - 4k}{2} = \frac{-k}{2} = \frac{4k - 3k}{2} = \frac{k}{2} \]

### Step 6: Calculate \( P_3F \)
And for \( P_3F \):
\[ P_3F = \frac{b - a}{2} = \frac{5k - 3k}{2} = \frac{2k}{2} = k \]

### Step 7: Substitute back into the equation
Now, substituting \( P_1D, P_2E, P_3F \) into the equation:
\[ \frac{k}{2} + \frac{k}{2} + k = 42 \]
This simplifies to:
\[ \frac{k}{2} + \frac{k}{2} + \frac{2k}{2} = 42 \]
\[ \frac{4k}{2} = 42 \]
\[ 2k = 42 \]
\[ k = 21 \]

### Step 8: Calculate \( a + b + c \)
Now that we have \( k \), we can find \( a + b + c \):
\[ a + b + c = 3k + 5k + 4k = 12k \]
Substituting \( k = 21 \):
\[ a + b + c = 12 \times 21 = 252 \]

### Final Answer
Thus, the value of \( a + b + c \) is \( \boxed{252} \).