Math Problem Statement

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

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?

Solution

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

Given:

  1. Point PP divides ABAB in the ratio 1:21:2. This means AP:PB=1:2AP:PB = 1:2.
  2. Lines BCBC and PDPD are parallel.

Problem: A) Find the ratio AD:CDAD:CD.

B) If AD=5cmAD = 5 \, \text{cm}, determine ACAC.

Part A: Finding AD:CDAD:CD

Since BCPDBC \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:

ADCD=APPB\frac{AD}{CD} = \frac{AP}{PB}

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

ADCD=12\frac{AD}{CD} = \frac{1}{2}

Thus, the ratio AD:CD=1:2AD:CD = 1:2.

Part B: Finding ACAC when AD=5cmAD = 5 \, \text{cm}

Since AD:CD=1:2AD:CD = 1:2, we can express CDCD in terms of ADAD:

Let CD=2xCD = 2x and AD=xAD = x. From the ratio 1:21:2, if AD=5cmAD = 5 \, \text{cm}, then:

x=5cmx = 5 \, \text{cm}

So, CD=2×5cm=10cmCD = 2 \times 5 \, \text{cm} = 10 \, \text{cm}.

Now, to find ACAC:

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

Final Answers:

A) AD:CD=1:2AD:CD = 1:2.

B) If AD=5cmAD = 5 \, \text{cm}, then AC=15cmAC = 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:PBAP: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.

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Math Problem Analysis

Mathematical Concepts

Ratios
Basic Proportionality Theorem
Parallel Lines

Formulas

AD/CD = AP/PB
AC = AD + CD

Theorems

Basic Proportionality Theorem (Thales' theorem)

Suitable Grade Level

Grades 8-10