Question

In $△ABC$, the bisector of $\angle A$ intersects $\overline{BC}$ at a point $D$. Then:

• A. $BD\times AC=BC\times AB$
• B. $BD\times AB=DC\times AC$
• C. $AC\times AB=DC\times BC$
• D. $BD\times AC=DC\times AB$

Answer 1

Answer: (D)

$BD\times AC=DC\times AB$

In $△ABC$, the bisector of $\angle A$ intersects $\overline{B}C$ at $D$.

$\therefore$ $AD$ bisects $BC$

$\therefore BD=DC$ ........ $\left(i\right)$

In triangles, $ABD$ and $ADC$

$BD=CD$ ......... From $\left(i\right)$

$\angle BAD=\angle DAC$ ....... (Given)

$AD=AD$ ........... (Common side)

$\therefore$ $△ABD\cong △ADC$ ........ [By S.A.S criterion]

$⟹AB=AC$

Thus, $BD\times AC=DC\times AB$