In the diagram, \(BC\) is parallel to \(DE\). \(AB = 5\) cm, \(BD = 7.5\) cm and \(BC = 6\) cm. Calculate the length of \(DE\).
Question 1 Report
In the diagram, \(BC\) is parallel to \(DE\).
\(AB = 5\) cm, \(BD = 7.5\) cm and \(BC = 6\) cm.
Calculate the length of \(DE\).
Answer Details
When \( BC \) is parallel to \( DE \), the line \( BC \) cuts triangle \( ADE \) so that angle \( ABC \) equals angle \( ADE \) and angle \( ACB \) equals angle \( AED \) (corresponding angles on parallel lines), and angle \( A \) is shared. Three equal pairs of angles means triangle \( ABC \) is similar to triangle \( ADE \).
The critical point is that the enlargement compares \( AB \) with the whole side \( AD \), not with the part \( BD \):
- \( AD = AB + BD = 5 + 7.5 = 12.5 \) cm.
- \( \text{scale factor} = \dfrac{AD}{AB} = \dfrac{12.5}{5} = 2.5 \) [M1]
- \( DE = BC \times 2.5 = 6 \times 2.5 = 15 \)
The length of \( DE \) is \( 15 \) cm. [A1]
The most frequent mistake is using \( 7.5 \div 5 = 1.5 \) and answering \( 9 \) cm. That compares a part of the large triangle with the small triangle, which are not corresponding sides. Always rebuild the full side of the larger triangle first.
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