Question:
In the given figure, DE $\parallel$ BC. If AD : AB = 1 : 3 and AE = 2.5 cm, then AC equals
In the given figure, DE $\parallel$ BC. If AD : AB = 1 : 3 and AE = 2.5 cm, then AC equals
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Always look at the given ratio carefully.
Sometimes the question gives \(AD : DB\) instead of \(AD : AB\).
Since it is given as \(AD : AB = 1 : 3\), the total side \(AC\) is simply 3 times the smaller part \(AE\).
Calculating \(3 \times 2.5 = 7.5\text{ cm}\) takes only a few seconds!
Sometimes the question gives \(AD : DB\) instead of \(AD : AB\).
Since it is given as \(AD : AB = 1 : 3\), the total side \(AC\) is simply 3 times the smaller part \(AE\).
Calculating \(3 \times 2.5 = 7.5\text{ cm}\) takes only a few seconds!
Updated On: Jun 25, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
This question is based on the topic of Similarity of Triangles and the Basic Proportionality Theorem (Thales's Theorem) from geometry.
We are given a triangle \(ABC\) where a line segment \(DE\) is drawn parallel to the base \(BC\), intersecting \(AB\) at \(D\) and \(AC\) at \(E\).
We are given the ratio of the segments \(AD : AB = 1 : 3\) and the length of segment \(AE = 2.5\text{ cm}\).
We need to determine the total length of the side \(AC\).
Step 2: Key Formula or Approach:
When a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides in the same ratio.
By the Basic Proportionality Theorem, or by using the similarity of triangles \(\triangle ADE\) and \(\triangle ABC\):
Since \(DE \parallel BC\), we have \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\) (corresponding angles).
Thus, by AA similarity, \(\triangle ADE \sim \triangle ABC\).
The ratio of corresponding sides of similar triangles is equal: \[ \frac{AD}{AB} = \frac{AE}{AC} \]
Step 3: Detailed Explanation:
1. Identify the given values from the problem statement:
- Ratio of sides: \(\frac{AD}{AB} = \frac{1}{3}\)
- Length of segment \(AE = 2.5\text{ cm}\)
2. Apply the similarity ratio established in Step 2: \[ \frac{AD}{AB} = \frac{AE}{AC} \] 3. Substitute the known values into the equation: \[ \frac{1}{3} = \frac{2.5}{AC} \] 4. Solve for the unknown length \(AC\) by cross-multiplying the terms: \[ AC = 3 \times 2.5 \] \[ AC = 7.5\text{ cm} \]
Step 4: Final Answer:
The length of the side \(AC\) is calculated to be \(7.5\text{ cm}\).
Therefore, the correct option is (A).
This question is based on the topic of Similarity of Triangles and the Basic Proportionality Theorem (Thales's Theorem) from geometry.
We are given a triangle \(ABC\) where a line segment \(DE\) is drawn parallel to the base \(BC\), intersecting \(AB\) at \(D\) and \(AC\) at \(E\).
We are given the ratio of the segments \(AD : AB = 1 : 3\) and the length of segment \(AE = 2.5\text{ cm}\).
We need to determine the total length of the side \(AC\).
Step 2: Key Formula or Approach:
When a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides the two sides in the same ratio.
By the Basic Proportionality Theorem, or by using the similarity of triangles \(\triangle ADE\) and \(\triangle ABC\):
Since \(DE \parallel BC\), we have \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\) (corresponding angles).
Thus, by AA similarity, \(\triangle ADE \sim \triangle ABC\).
The ratio of corresponding sides of similar triangles is equal: \[ \frac{AD}{AB} = \frac{AE}{AC} \]
Step 3: Detailed Explanation:
1. Identify the given values from the problem statement:
- Ratio of sides: \(\frac{AD}{AB} = \frac{1}{3}\)
- Length of segment \(AE = 2.5\text{ cm}\)
2. Apply the similarity ratio established in Step 2: \[ \frac{AD}{AB} = \frac{AE}{AC} \] 3. Substitute the known values into the equation: \[ \frac{1}{3} = \frac{2.5}{AC} \] 4. Solve for the unknown length \(AC\) by cross-multiplying the terms: \[ AC = 3 \times 2.5 \] \[ AC = 7.5\text{ cm} \]
Step 4: Final Answer:
The length of the side \(AC\) is calculated to be \(7.5\text{ cm}\).
Therefore, the correct option is (A).
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