Question:
In the given figure, \(DE \parallel BC\). If \(AD : AB = 1 : 3\) and \(AE = 2.5\text{ cm}\), then \(AC\) equals
In the given figure, \(DE \parallel BC\). If \(AD : AB = 1 : 3\) and \(AE = 2.5\text{ cm}\), then \(AC\) equals
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Remember that the ratio of the parts to the whole is constant across both sides of the triangle when a line is parallel to the base.
If the left side has a part-to-whole ratio of 1 to 3, the right side must also have a part-to-whole ratio of 1 to 3.
Thus, \(AC = 3 \times AE = 3 \times 2.5 = 7.5\text{ cm}\).
If the left side has a part-to-whole ratio of 1 to 3, the right side must also have a part-to-whole ratio of 1 to 3.
Thus, \(AC = 3 \times AE = 3 \times 2.5 = 7.5\text{ cm}\).
Updated On: Jun 25, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We are given \(\triangle ABC\) where a line segment \(DE\) is parallel to the base \(BC\), with \(D\) lying on \(AD\) and \(E\) lying on \(AC\).
We are given the ratio \(\frac{AD}{AB} = \frac{1}{3}\) 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:
According to Thales's Theorem (Basic Proportionality Theorem) and the properties of similar triangles:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the smaller triangle formed is similar to the larger triangle.
In this case, since \(DE \parallel BC\), we have \(\triangle ADE \sim \triangle ABC\) by AA similarity criterion (since \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\) as corresponding angles).
Since corresponding sides of similar triangles are in the same ratio:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
Step 3: Detailed Explanation:
• From the problem, the given ratio is:
\[ \frac{AD}{AB} = \frac{1}{3} \]
• The length of the corresponding segment on the other side is given as:
\[ AE = 2.5\text{ cm} \]
• Set up the proportion from the similarity relationship:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
• Substitute the known values into this proportion:
\[ \frac{1}{3} = \frac{2.5}{AC} \]
• To solve for \(AC\), cross-multiply:
\[ 1 \times AC = 3 \times 2.5 \]
\[ AC = 7.5\text{ cm} \]
Step 4: Final Answer:
The length of \(AC\) is 7.5 cm. This matches option (A).
We are given \(\triangle ABC\) where a line segment \(DE\) is parallel to the base \(BC\), with \(D\) lying on \(AD\) and \(E\) lying on \(AC\).
We are given the ratio \(\frac{AD}{AB} = \frac{1}{3}\) 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:
According to Thales's Theorem (Basic Proportionality Theorem) and the properties of similar triangles:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the smaller triangle formed is similar to the larger triangle.
In this case, since \(DE \parallel BC\), we have \(\triangle ADE \sim \triangle ABC\) by AA similarity criterion (since \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\) as corresponding angles).
Since corresponding sides of similar triangles are in the same ratio:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
Step 3: Detailed Explanation:
• From the problem, the given ratio is:
\[ \frac{AD}{AB} = \frac{1}{3} \]
• The length of the corresponding segment on the other side is given as:
\[ AE = 2.5\text{ cm} \]
• Set up the proportion from the similarity relationship:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
• Substitute the known values into this proportion:
\[ \frac{1}{3} = \frac{2.5}{AC} \]
• To solve for \(AC\), cross-multiply:
\[ 1 \times AC = 3 \times 2.5 \]
\[ AC = 7.5\text{ cm} \]
Step 4: Final Answer:
The length of \(AC\) is 7.5 cm. This matches option (A).
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