In \(\Delta\) ABC, AD is a median. X is a point on AD such that AX : XD = 2 : 3. BX is extended so that it intersects AC at Y. Prove that BX = 4 XY.
Show Hint
Drawing a line parallel to the bisector or the transversal from the midpoint of the base is a standard construction trick in triangle geometry.
It instantly creates midpoints and similar triangles, converting a complex coordinate geometry problem into a simple ratio verification.
Step 1: Understanding the Question:
The topic of this question is Triangles (Basic Proportionality Theorem and similarity applications).
We are given a triangle \( ABC \) with median \( AD \), which means \( D \) is the midpoint of side \( BC \).
Point \( X \) lies on the median \( AD \) such that \( AX : XD = 2 : 3 \).
The segment \( BX \) is extended to meet side \( AC \) at \( Y \). We need to prove that \( BX = 4XY \).
Step 2: Key Formula or Approach:
- Introduce a parallel line construction: Draw a line from \( D \) parallel to \( BY \) that intersects side \( AC \) at point \( E \).
- Apply the Basic Proportionality Theorem (BPT) in \( \Delta BCY \) and \( \Delta ADE \) to set up ratios.
Step 3: Detailed Explanation:
1. Construction:
Draw a line segment \( DE \) parallel to \( BY \) (or \( XY \)) meeting \( AC \) at point \( E \):
\[ DE \parallel BY \]
2. Analyze \(\Delta BCY\):
In triangle \( \Delta BCY \), we have \( DE \parallel BY \).
Since \( AD \) is the median of \( \Delta ABC \), \( D \) is the midpoint of \( BC \).
By the Converse of the Midpoint Theorem, since \( D \) is the midpoint of \( BC \) and \( DE \parallel BY \), the point \( E \) must be the midpoint of \( CY \):
\[ YE = EC \quad \text{(Equation 1)} \]
Also, by the Midpoint Theorem, the length of \( DE \) is half of the base \( BY \):
\[ DE = \frac{1}{2} BY \implies BY = 2 DE \quad \text{(Equation 2)} \]
3. Analyze \(\Delta ADE\):
In triangle \( \Delta ADE \), \( XY \) is parallel to \( DE \) (since \( BY \parallel DE \)).
By the Basic Proportionality Theorem (or similarity of \( \Delta AXY \sim \Delta ADE \)):
\[ \frac{XY}{DE} = \frac{AX}{AD} \quad \text{(Equation 3)} \]
4. Calculate ratios using given information:
We are given the ratio:
\[ \frac{AX}{XD} = \frac{2}{3} \]
Thus, the ratio of \( AX \) to the entire median \( AD \) is:
\[ \frac{AX}{AD} = \frac{AX}{AX + XD} = \frac{2}{2 + 3} = \frac{2}{5} \]
5. Substitute this ratio into Equation 3:
\[ \frac{XY}{DE} = \frac{2}{5} \implies DE = \frac{5}{2} XY \quad \text{(Equation 4)} \]
6. Substitute Equation 4 into Equation 2:
\[ BY = 2 \left(\frac{5}{2} XY\right) \]
\[ BY = 5 XY \]
7. Express \( BY \) in terms of its segments \( BX \) and \( XY \):
\[ BY = BX + XY \]
Substitute this into the equation:
\[ BX + XY = 5 XY \]
\[ BX = 5 XY - XY \]
\[ BX = 4 XY \]
This completes the formal geometric proof.
Step 4: Final Answer:
Hence, it is proved that \(BX = 4XY\).
