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Similarity — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsSimilarity4 Marks
Question
The given figure shows a parallelogram ABCD. E is a point in AD and CE produced meets BA produced at point F. If AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.
✓

Answer

$AF = 8$ cm and $AB = 12\ cm$
So, $FB = 20 cm$
In $\triangle DEC$ and $\triangle EAF$
$\angle DEC = \angle EAF ....$(vertically opposite angles)
$\angle EDC = \angle EAF ....$(alternate angles)
So, $\triangle DEC \sim \triangle AEF ...$(AA criterion for similarity)
$\Rightarrow \frac{D E}{A E}=\frac{E C}{E F}=\frac{D C}{A F}$
$\Rightarrow \frac{D E}{A E}=\frac{D C}{A F}$
$\Rightarrow \frac{D E}{A E}=\frac{A B}{A F}$
$\Rightarrow \frac{D E}{4}=\frac{12}{8}$
$\Rightarrow$DE = 6 cm
So, $AD = AE + ED = 4 + 6 = 10 cm$
Perimeter of the parallelogram $ABCD$
$= AB + BC + CD + AD$
$= 12 + 10 + 12 + 10$
$= 44 cm$

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