Question:
In \(\triangle ABC\), D lies on AB and E lies on AC such that AD = 2 cm, DB = 3 cm, AE = 4 cm and EC = 6 cm. By the converse of the Basic Proportionality Theorem, ________.
In \(\triangle ABC\), D lies on AB and E lies on AC such that AD = 2 cm, DB = 3 cm, AE = 4 cm and EC = 6 cm. By the converse of the Basic Proportionality Theorem, ________.
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Whenever you are given BPT-like segment values, quickly compute and compare the two ratios.
If they are equal, the lines are parallel.
Here, \(2/3\) is obviously equal to \(4/6\), leading directly to \(DE \parallel BC\).
If they are equal, the lines are parallel.
Here, \(2/3\) is obviously equal to \(4/6\), leading directly to \(DE \parallel BC\).
Updated On: Jul 13, 2026
- DE \(\perp\) BC
- DE \(\parallel\) AB
- DE bisects BC
- DE \(\parallel\) BC
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
This question is from Geometry, focusing on the application of the Converse of the Basic Proportionality Theorem (BPT).
We are given the lengths of the segments on sides \(AB\) and \(AC\) created by a line segment \(DE\). We need to determine the geometric relationship between \(DE\) and \(BC\).
Step 2: Key Formula or Approach:
The Converse of the Basic Proportionality Theorem states:
If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side of the triangle.
Thus, if:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]
then:
\[ DE \parallel BC \]
Step 3: Detailed Explanation:
We are given the following lengths:
\[ AD = 2\text{ cm} \]
\[ DB = 3\text{ cm} \]
\[ AE = 4\text{ cm} \]
\[ EC = 6\text{ cm} \]
Let us calculate the ratio of the segments on the left side (side \(AB\)):
\[ \frac{AD}{DB} = \frac{2}{3} \]
Next, let us calculate the ratio of the segments on the right side (side \(AC\)):
\[ \frac{AE}{EC} = \frac{4}{6} \]
Simplify the fraction \(\frac{4}{6}\) by dividing the numerator and denominator by their greatest common divisor, which is 2:
\[ \frac{AE}{EC} = \frac{2}{3} \]
Comparing the two ratios:
\[ \frac{AD}{DB} = \frac{AE}{EC} = \frac{2}{3} \]
Since the line segment \(DE\) divides both sides \(AB\) and \(AC\) in the same ratio, the condition for the converse of BPT is fully satisfied.
Therefore, the line segment \(DE\) is parallel to the third side \(BC\).
Step 4: Final Answer:
By the converse of BPT, DE \(\parallel\) BC.
This question is from Geometry, focusing on the application of the Converse of the Basic Proportionality Theorem (BPT).
We are given the lengths of the segments on sides \(AB\) and \(AC\) created by a line segment \(DE\). We need to determine the geometric relationship between \(DE\) and \(BC\).
Step 2: Key Formula or Approach:
The Converse of the Basic Proportionality Theorem states:
If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side of the triangle.
Thus, if:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]
then:
\[ DE \parallel BC \]
Step 3: Detailed Explanation:
We are given the following lengths:
\[ AD = 2\text{ cm} \]
\[ DB = 3\text{ cm} \]
\[ AE = 4\text{ cm} \]
\[ EC = 6\text{ cm} \]
Let us calculate the ratio of the segments on the left side (side \(AB\)):
\[ \frac{AD}{DB} = \frac{2}{3} \]
Next, let us calculate the ratio of the segments on the right side (side \(AC\)):
\[ \frac{AE}{EC} = \frac{4}{6} \]
Simplify the fraction \(\frac{4}{6}\) by dividing the numerator and denominator by their greatest common divisor, which is 2:
\[ \frac{AE}{EC} = \frac{2}{3} \]
Comparing the two ratios:
\[ \frac{AD}{DB} = \frac{AE}{EC} = \frac{2}{3} \]
Since the line segment \(DE\) divides both sides \(AB\) and \(AC\) in the same ratio, the condition for the converse of BPT is fully satisfied.
Therefore, the line segment \(DE\) is parallel to the third side \(BC\).
Step 4: Final Answer:
By the converse of BPT, DE \(\parallel\) BC.
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