Question

In the given figure, $A B ∥ D E$ and $A C ∥ D F$ . Show that $△ A BC \sim △ D EF$ . If $BC = 10$ cm, $EB = CF = 5$ cm and $A B = 7$ cm, then find the length DE.

Approach

Given $A B ∥ D E$ and $A C ∥ D F$ , we have $∠ D EF = ∠ A BC$ and $∠ D FE = ∠ A CB$ , so $△ A BC \sim △ D EF$ by AA similarity. Using proportionality: $\frac{BC}{EF} = \frac{A B}{D E}$ , so $\frac{10}{20} = \frac{7}{D E}$ , giving $D E = 14$ cm.

Step-by-step working

1
$A B ∥ D E \Rightarrow ∠ D EF = ∠ A BC$ ; $A C ∥ D F \Rightarrow ∠ D FE = ∠ A CB$ . Hence $△ A BC \sim △ D EF$ by AA.
1 mark
2
$\frac{BC}{EF} = \frac{A B}{D E} \Rightarrow \frac{10}{20} = \frac{7}{D E} \Rightarrow D E = 14$ cm
0.5 marks

Concepts used

similar trianglesparallel linesprove similarityfind length

Approach

Step-by-step working

Concepts used

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