Question

One side of a given triangle is $18$ inches. Inside the triangle a line segment is drawn parallel to this side, cutting off a triangle whose area is two-thirds that of the given triangle. Find the length of this segment in inches.

• A. $12$
• B. $6\sqrt{6}$
• C. $9\sqrt{2}$
• D. $6\sqrt{3}$
• E. $9$

Answer 1

The correct option is B $6\sqrt{6}$

Let the length be $a$
Let us assume that the height of original triangle be $p$ and height of triangle after cut is $q$
By similarity, we get $\frac{a}{18}=\frac{q}{p}$
By given relation of areas, we get $\frac{1}{2}\times a\times q=\frac{1}{2}\times 18\times p\times \frac{2}{3}$
Which gives $a^2=324\times \frac{2}{3}=216$ $\Rightarrow a=6\sqrt{6}$