Question

A set of triangles is formed by joining the midpoints of the larger triangles. If the area $△ABC$ is $128$, then the area of $△DEF$, the smallest triangle formed, is:

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $4$
• E. $1$

Answer 1

Answer: (C)

The segments joining the midpoints of two sides of a triangle is half as long as the third side. $\overline{DE}$ is the consequence of the $4^{th}$ set of midpoints so,

$\frac{DE}{BC}={\left(\frac{1}{2}\right)}^4=\frac{1}{16}$

The ratio of the areas is the square of the ratio of corresponding sides,

$\therefore \frac{Area△FDE}{Area△ABC}={\left(\frac{1}{16}\right)}^2=\frac{1}{256}$

Area of $△DEF$ is given by:

$Area△DEF=\frac{Area△ABC}{256}=\frac{128}{256}=\frac{1}{2}$