Question

In the figure, AC = DE = 7 cm. If BD = 4 cm, then CF equals

• A. 2 cm
• B. 3 cm
• C. 4 cm
• D. 5 cm

Answer 1

The correct option is C

4 cm

Construction: Extend ED to G, where it cuts AB. Similarly, extend AC to cut EF at H.
$\angle EDF=\angle BCA$ [Given]
Given, $AB\parallel EF$.
$\angle DEF=\angle DGB\cdots$ (1)
(Alternate interior angles)
Since $AC\parallel DE$, we must have
$\angle DGB=\angle BAC\cdots$ (2) (Corresponding angles)
From (1) and (2),
$\angle DEF=\angle BAC$
In $\Delta ABC$ and $\Delta EFD$,
$\angle ACB=\angle EDF$ [Given]
$\angle BAC=\angle DEF$ [Proven]
AC = DE [Given]
$⟹\Delta ABC\cong \Delta EFD$ (AAS congruency criteria)
So, BC = DF [CPCT]
$⟹$ BC - DC = DF - DC
$⟹$ BD = CF
$⟹$ CF = 4 cm [$\because$ BD = 4 cm (given)]