ABCD is a parallelogram. E is a point on extended side AB such that AB = BE. By joining D to E it intersects side BC at P. Find length of PC if BC = 15 cm.

Answer

To solve the problem step by step, we will use the properties of a parallelogram and the midpoint theorem.

### Step 1: Understand the Geometry
We have a parallelogram ABCD, where E is a point on the extended line of AB such that AB = BE. This means that E is twice the distance from A to B.

### Step 2: Identify the Midpoint
Since E is on the extended line of AB and AB = BE, we can conclude that B is the midpoint of AE. Therefore, we can denote:
- AE = AB + BE = 2 * AB

### Step 3: Apply the Midpoint Theorem
In triangle AED, by the midpoint theorem, we know that if B is the midpoint of AE, then:
\[ BP = \frac{1}{2} AD \]
This is because AD is parallel to BP and BC.

### Step 4: Relate AD and BC
Since ABCD is a parallelogram, we know that:
\[ AD = BC \]
Thus, we can substitute BC for AD in our earlier equation:
\[ BP = \frac{1}{2} BC \]

### Step 5: Substitute the Value of BC
We are given that:
\[ BC = 15 \, \text{cm} \]
Substituting this value into the equation gives us:
\[ BP = \frac{1}{2} \times 15 = 7.5 \, \text{cm} \]

### Step 6: Find Length of PC
Since P divides BC into two segments, we can find PC by subtracting BP from BC:
\[ PC = BC - BP \]
\[ PC = 15 - 7.5 = 7.5 \, \text{cm} \]

### Final Answer
The length of PC is:
\[ \text{PC} = 7.5 \, \text{cm} \]

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