Three sides of a triangle ABC are `a, b, c. a=4700 cm, b=4935 cm and c=6815 cm`. The internal bisector of ZA meets BC at P, and the bisector passes through incentre O. What is ratio of `PO:OA` ?

Options

A. `3:2`
B. `2:3`
C. `2:5`
D. can't be determined

Answer

To find the ratio of \( PO:OA \) in triangle \( ABC \) with sides \( a = 4700 \, \text{cm} \), \( b = 4935 \, \text{cm} \), and \( c = 6815 \, \text{cm} \), we can follow these steps:

### Step 1: Find the ratio of the sides
We start by simplifying the sides of the triangle. We can divide each side by 5 to make the numbers smaller:

\[ \frac{a}{5} = \frac{4700}{5} = 940, \quad \frac{b}{5} = \frac{4935}{5} = 987, \quad \frac{c}{5} = \frac{6815}{5} = 1363 \]

Next, we can further simplify by dividing each of these values by their greatest common divisor. In this case, we can divide by 47:

\[ \frac{940}{47} = 20, \quad \frac{987}{47} = 21, \quad \frac{1363}{47} = 29 \]

Thus, the simplified ratio of the sides \( a:b:c \) is \( 20:21:29 \).

### Step 2: Identify the type of triangle
The ratio \( 20:21:29 \) suggests that this triangle is a right triangle. We can confirm this using the Pythagorean theorem:

\[ 20^2 + 21^2 = 400 + 441 = 841 = 29^2 \]

This confirms that triangle \( ABC \) is indeed a right triangle.

### Step 3: Calculate the inradius
The inradius \( r \) of a triangle can be calculated using the formula:

\[ r = \frac{a + b - c}{2} \]

Substituting the values:

\[ r = \frac{20 + 21 - 29}{2} = \frac{12}{2} = 6 \, \text{cm} \]

### Step 4: Set up the triangle and points
In triangle \( ABC \):
- Let \( O \) be the incenter.
- Let \( P \) be the point where the internal bisector of angle \( A \) meets side \( BC \).

### Step 5: Use similar triangles to find the ratio
In triangles \( AMO \) and \( ACP \) (where \( M \) is the point where the inradius meets \( BC \)), we can establish that:

\[ \frac{AM}{MC} = \frac{AO}{OP} \]

Given that:
- \( AM = 21 - 6 = 15 \)
- \( MC = 6 \)

Thus, we have:

\[ \frac{15}{6} = \frac{AO}{OP} \]

This simplifies to:

\[ \frac{5}{2} = \frac{AO}{OP} \]

### Step 6: Find the desired ratio
To find the ratio \( PO:OA \), we take the reciprocal of \( \frac{AO}{OP} \):

\[ PO:OA = \frac{2}{5} \]

### Final Answer
Thus, the ratio \( PO:OA \) is \( \frac{2}{5} \).

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