Question

In the figure above, $AB=6$ and $BC=8$. Find the length of segment $\overline{BD}$.

• A. $2$
• B. $\frac{12}{5}$
• C. $4$
• D. $\frac{24}{5}$
• E. $6$

Answer 1

The correct option is D $\frac{24}{5}$

As given $AB=6$ and $BC=8$
Since $ABC$ is a right triangle, the length of segment $AC$, according to pythagorean theorem;
$A{C}^2=A{B}^2+B{C}^2$
$\Rightarrow A{C}^2=6^2+8^2$
$\Rightarrow AC=\sqrt{6^2+8^2}$
$\Rightarrow AC=10$
The area of triangle $ABC$ when base is $AB$ and height is $BC$,
$=\frac{1}{2}\times$ base $\times$ height
$=\frac{1}{2}\times AB\times BC$
$=\frac{1}{2}\times 6\times 8$
$=24$
Again, the area of triangle $ABC$ when base is $AC$ and height is $BD$ which is same as the area of triangle $ABC$ when base is $AB$ and height is $BC$,
$24=\frac{1}{2}\times AC\times BD$
$\Rightarrow 24=\frac{1}{2}\times 10\times BD$
$\Rightarrow 24=5\times BD$
$\Rightarrow BD=\frac{24}{5}$
Hence, option D is correct.