ABCD is a parallelogram AB is divided at P and CD at Q so that AP:PB = 3:2 and CQ:QD = 4:1 if PQ meets AC at R then AR =

Options

A. `2/7` AC
B. `3/7`AC
C. `4/7`AC
D. `5/7`AC

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given ratios
We are given that in parallelogram ABCD, the segment AB is divided at point P such that the ratio AP:PB = 3:2. Similarly, the segment CD is divided at point Q such that the ratio CQ:QD = 4:1.

### Step 2: Assign variables based on the ratios
Let:
- AP = 3x
- PB = 2x
- CQ = 4y
- QD = y

### Step 3: Calculate the lengths of AB and CD
Since AB = AP + PB, we have:
AB = 3x + 2x = 5x

Similarly, for CD:
CD = CQ + QD = 4y + y = 5y

### Step 4: Establish the relationship between x and y
Since ABCD is a parallelogram, opposite sides are equal. Therefore:
AB = CD
5x = 5y
This implies that x = y.

### Step 5: Substitute y with x
Now, we can express CQ and QD in terms of x:
- CQ = 4x
- QD = x

### Step 6: Identify similar triangles
When line PQ intersects AC at point R, triangles ARP and CRQ are formed. We can say that these triangles are similar.

### Step 7: Set up the proportion based on the similarity of triangles
From the similarity of triangles ARP and CRQ, we can write:
\[ \frac{AP}{CQ} = \frac{AR}{CR} \]
Substituting the values we have:
\[ \frac{3x}{4x} = \frac{AR}{CR} \]
This simplifies to:
\[ \frac{3}{4} = \frac{AR}{CR} \]

### Step 8: Express CR in terms of AR
Let AR = 3k and CR = 4k for some k. Then:
\[ AC = AR + CR = 3k + 4k = 7k \]

### Step 9: Find AR in terms of AC
Now we can express AR as a fraction of AC:
\[ \frac{AR}{AC} = \frac{3k}{7k} = \frac{3}{7} \]
Thus, we have:
\[ AR = \frac{3}{7} \times AC \]

### Conclusion
The length of AR is \(\frac{3}{7}\) of the length of AC.