Question

In $\Delta ABC,\angle ABC=\angle DAC$, AB = 8 cm, AC = 4 cm, AD = 5 cm. [4 MARKS]
i) Prove that $\Delta ACD$ is similar to $\Delta BCA$
ii) Find BC and CD
iii) Find area of $\Delta ACD$ : area of $\Delta ABC$

Answer 1

Applying theorems: 2 Marks
Calculation: 2 Marks

Given: In $\Delta ABC$,
$\angle ABC=\angle DAC$,
AB = 8 cm, AC = 4 cm, AD = 5 cm
i) Considering $\Delta ACD$ and $\Delta BCA$,
$\angle DAC=\angle ABC$ [Given]
$\angle C=\angle C$ [Common]
$\therefore \Delta ACD\sim \Delta BCA$ [By AA axiom of similarity]

ii) $\Delta ACD\sim \Delta BCA$ [Proved above]
$\Rightarrow \frac{AC}{BC}=\frac{CD}{CA}=\frac{AD}{BA}$
$\Rightarrow \frac{4}{BC}=\frac{CD}{4}=\frac{5}{8}$
$\therefore BC=6.4cm;CD=2.5cm$

iii) $\Delta ACD\sim \Delta ABC$
As the ratio of the areas of two similar triangle is equal to the ratio of the squares of any two corresponding sides.
$\therefore \frac{Areaof\Delta ACD}{Areaof\Delta ABC}=\frac{A{D}^2}{B{A}^2}=\frac{25}{64}$
Hence, area of $\Delta ACD$ : Area of $\Delta ABC$ = 25 : 64