Question:
In the given figure \(\Delta ABC\) is shown, in which \(DE \parallel BC\). If \(AD = 5\text{ cm}\), \(DB = 2\cdot5\text{ cm}\) and \(DE = 8\text{ cm}\), then the length of \(BC\) is :
In the given figure \(\Delta ABC\) is shown, in which \(DE \parallel BC\). If \(AD = 5\text{ cm}\), \(DB = 2\cdot5\text{ cm}\) and \(DE = 8\text{ cm}\), then the length of \(BC\) is :
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Do not confuse Thales' Theorem (Basic Proportionality Theorem) with Triangle Similarity.
BPT states \(\frac{AD}{DB} = \frac{AE}{EC}\), which involves only segments on the sides.
When calculating lengths of parallel parallel bases like \(DE\) and \(BC\), you must use similar triangles: \(\frac{AD}{AB} = \frac{DE}{BC}\).
BPT states \(\frac{AD}{DB} = \frac{AE}{EC}\), which involves only segments on the sides.
When calculating lengths of parallel parallel bases like \(DE\) and \(BC\), you must use similar triangles: \(\frac{AD}{AB} = \frac{DE}{BC}\).
Updated On: Jul 7, 2026
- 10 cm
- 6 cm
- 12 cm
- 75 cm
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
The question asks us to find the length of side \(BC\) in \(\Delta ABC\), where a line segment \(DE\) is parallel to \(BC\) with given segment lengths.
Step 2: Key Formula or Approach:
Since \(DE \parallel BC\):
1. The corresponding angles are equal, i.e., \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\).
2. By Angle-Angle (AA) similarity, \(\Delta ADE \sim \Delta ABC\).
3. For similar triangles, the ratio of corresponding sides is equal:
\[ \frac{AD}{AB} = \frac{DE}{BC} \]
Note that \(AB = AD + DB\).
Step 3: Detailed Explanation:
1. Identify the given values from the problem statement:
\[ AD = 5\text{ cm}, \quad DB = 2.5\text{ cm}, \quad DE = 8\text{ cm} \]
2. Calculate the total length of side \(AB\):
\[ AB = AD + DB = 5 + 2.5 = 7.5\text{ cm} \]
3. Establish the similarity ratio between the similar triangles \(\Delta ADE\) and \(\Delta ABC\):
\[ \frac{AD}{AB} = \frac{DE}{BC} \]
4. Substitute the known values into the ratio:
\[ \frac{5}{7.5} = \frac{8}{BC} \]
5. Simplify the fraction on the left-hand side:
\[ \frac{5}{7.5} = \frac{5}{\frac{15}{2}} = \frac{10}{15} = \frac{2}{3} \]
6. Now set up the simplified proportion:
\[ \frac{2}{3} = \frac{8}{BC} \]
7. Solve for \(BC\) by cross-multiplying:
\[ 2 \times BC = 3 \times 8 \]
\[ 2 \times BC = 24 \]
\[ BC = 12\text{ cm} \]
Step 4: Final Answer:
Hence, the correct option is (C).
The question asks us to find the length of side \(BC\) in \(\Delta ABC\), where a line segment \(DE\) is parallel to \(BC\) with given segment lengths.
Step 2: Key Formula or Approach:
Since \(DE \parallel BC\):
1. The corresponding angles are equal, i.e., \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\).
2. By Angle-Angle (AA) similarity, \(\Delta ADE \sim \Delta ABC\).
3. For similar triangles, the ratio of corresponding sides is equal:
\[ \frac{AD}{AB} = \frac{DE}{BC} \]
Note that \(AB = AD + DB\).
Step 3: Detailed Explanation:
1. Identify the given values from the problem statement:
\[ AD = 5\text{ cm}, \quad DB = 2.5\text{ cm}, \quad DE = 8\text{ cm} \]
2. Calculate the total length of side \(AB\):
\[ AB = AD + DB = 5 + 2.5 = 7.5\text{ cm} \]
3. Establish the similarity ratio between the similar triangles \(\Delta ADE\) and \(\Delta ABC\):
\[ \frac{AD}{AB} = \frac{DE}{BC} \]
4. Substitute the known values into the ratio:
\[ \frac{5}{7.5} = \frac{8}{BC} \]
5. Simplify the fraction on the left-hand side:
\[ \frac{5}{7.5} = \frac{5}{\frac{15}{2}} = \frac{10}{15} = \frac{2}{3} \]
6. Now set up the simplified proportion:
\[ \frac{2}{3} = \frac{8}{BC} \]
7. Solve for \(BC\) by cross-multiplying:
\[ 2 \times BC = 3 \times 8 \]
\[ 2 \times BC = 24 \]
\[ BC = 12\text{ cm} \]
Step 4: Final Answer:
Hence, the correct option is (C).
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