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In the figure p divides AB in the ratio 1:2 The  lines BC and PD are parallel lines

A)what is AD:CD?

B) if AD=5cm then what is AC

To solve the problem, let's analyze the given figure and information:

**Given:**
1. Point \( P \) divides \( AB \) in the ratio \( 1:2 \). This means \( AP:PB = 1:2 \).
2. Lines \( BC \) and \( PD \) are parallel.

**Problem:**
A) Find the ratio \( AD:CD \).

B) If \( AD = 5 \, \text{cm} \), determine \( AC \).

### Part A: Finding \( AD:CD \)

Since \( BC \parallel PD \), by the Basic Proportionality Theorem (or Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally, we have:

\[
\frac{AD}{CD} = \frac{AP}{PB}
\]

Given that \( AP:PB = 1:2 \), we substitute this ratio:

\[
\frac{AD}{CD} = \frac{1}{2}
\]

Thus, the ratio \( AD:CD = 1:2 \).

### Part B: Finding \( AC \) when \( AD = 5 \, \text{cm} \)

Since \( AD:CD = 1:2 \), we can express \( CD \) in terms of \( AD \):

Let \( CD = 2x \) and \( AD = x \). From the ratio \( 1:2 \), if \( AD = 5 \, \text{cm} \), then:

\[
x = 5 \, \text{cm}
\]

So, \( CD = 2 \times 5 \, \text{cm} = 10 \, \text{cm} \).

Now, to find \( AC \):

\[
AC = AD + CD = 5 \, \text{cm} + 10 \, \text{cm} = 15 \, \text{cm}
\]

### Final Answers:
A) \( AD:CD = 1:2 \).

B) If \( AD = 5 \, \text{cm} \), then \( AC = 15 \, \text{cm} \).

Would you like a more detailed explanation or have any other questions?

Here are some related questions:
1. How does the Basic Proportionality Theorem work in other triangle configurations?
2. What would happen if the ratio \( AP:PB \) was different?
3. How can we prove that lines are parallel using triangle ratios?
4. How does the concept of similar triangles relate to the Basic Proportionality Theorem?
5. What are other applications of parallel lines in geometry problems?

**Tip:** When working with proportionality in triangles, remember that parallel lines play a crucial role in dividing the sides proportionally.

In the figure P divides AB in the ratio 1:2. The lines BC and PD are parallel lines.
A) What is AD:CD?
B) If AD = 5 cm, then what is AC?

Solve the geometric problem involving the ratio AD:CD and length AC using the Basic Proportionality Theorem (Thales' Theorem). Learn how to apply the ratio AP:PB = 1:2 when AD = 5 cm. Suitable for students in Grades 8-10.

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