Question

ABC is a triangle in which ∠ A = 90°, AN ⊥ BC, BC = 12 cm and AC = 5 cm. Find the ratio of the areas of ΔANC and ΔABC.

• A. 13 : 45
• B. 14 : 93
• C. 25 : 144
• D. 1 : 5

Answer 1

The correct option is C 25 : 144

In ΔANC and ΔABC,
⇒∠ACN = ∠ACB [Common]
⇒∠A = ∠ANC [each ${90}^{\circ }$]
ΔANC ∼ ΔBAC [AA similarity]

Since the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides, we have

$\frac{Ar\left(\Delta ANC\right)}{Ar\left(\Delta ABC\right)}=(\frac{AC}{BC}{)}^2$
$\frac{Ar\left(\Delta ANC\right)}{Ar\left(\Delta ABC\right)}=(\frac{5}{12}{)}^2$
$\frac{Ar\left(\Delta ANC\right)}{Ar\left(\Delta ABC\right)}=\frac{25}{144}$

Therefore,
$\frac{Ar\left(\Delta ANC\right)}{Ar\left(\Delta ABC\right)}=25:144$