Question

The sides of a triangle are 28 cm, 36 cm and 48 cm. Find the lengths of line segments into which the smallest side is divided by the bisector of the angle opposite to it

Answer 1

$\textbf{Given:}$

$\text{Sides of the given triangle are 28 cm, 36 cm and 48 cm}$

$\textbf{To find:}$

$\text{Lengths of line segments into which the smallest side}$

$\text{is divided by the bisector of the angle opposite to it}$

$\textbf{Solution:}$

$\textbf{Angle bisector theorem:}$

$\text{When vertical angle of a triangle is bisected,}$

$\text{the bisector divides the base into two segments}$

$\text{which have the ratio as the order of other two sides}$

$\text{Let \triangle ABC be given triagle with AB=28 cm, BC=36 cm, AC=48 cm}$

$\text{In the given triangle, length of the smallest side is AB}$

$\text{As per given data, the bisector divides \angle{C} which is opposite to the smallest side AB}$

$\text{Let CD be the bisector of \angle{C} }$

$\text{By angle bisector theorem,}$

$\dfrac{AD}{BD}=\dfrac{AC}{BC}$

$\dfrac{AD}{BD}=\dfrac{48}{36}$

$\implies\dfrac{AD}{BD}=\dfrac{4}{3}$

$AD=\dfrac{4}{7}{\times}AB$

$AD=\dfrac{4}{7}{\times}28$

$AD=4{\times}4$

$AD=16\,cm$

$BD=\dfrac{3}{7}{\times}AB$

$BD=\dfrac{3}{7}{\times}28$

$BD=3{\times}4$

$BD=12\,cm$

$\textbf{Answer:}$

$\textbf{Lengths of the line segments are 16 cm and 12 cm}$

Find more:

AD=15 and DC=20. If BD is the bisector of angle ABC, what is the perimeter of the triangle ABC?

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