Question

The sides of a triangle are $5$ cm, $6$ cm and $7$ cm. One more triangle is formed by joining the midpoints of the sides. The perimeter of the second triangle is:

• A. $18$ cm
• B. $12$ cm
• C. $9$ cm
• D. $6$ cm

Answer 1

The correct option is C $9$ cm

Let the $△ABC$ have sides $AB=5$cm,

$BC=6$ cm and $AC=7$cm.

Let the midpoints of the sides AB and AC be points D and E respectively.

$\therefore \frac{AD}{DB}=\frac{AE}{EC}$ ...By B.P.T

$\therefore \frac{AD+DB}{DB}=\frac{AE+EC}{EC}$ ....By Componendo

$\therefore \frac{AB}{DB}=\frac{AC}{EC}$ ......(1)

Also, $\angle BAC\cong \angle DAE$ ....(2)

$\therefore △ABC\sim △ADE$ ....SAS test of similarity

$\therefore \frac{AB}{AD}=\frac{BC}{DE}=\frac{AC}{AE}$ ....C.S.S.T

But $\frac{AB}{AD}=\frac{AB}{\frac{1}{2}AB}=\frac{1}{2}$

$\therefore \frac{BC}{DE}=\frac{1}{2}$

Perimeter $\left(△ADE\right)=AD+DE+AE$

$=\frac{1}{2}AB+\frac{1}{2}BC+\frac{1}{2}AC$

$=\frac{1}{2}(AB+BC+AC)$

$=\frac{1}{2}\times 18=9$ cm.

So, option C is correct.