Question

In the figure B= 90 degrees, AD=15 and DC=20. If BD is the bisector of angle ABC, what is the perimeter of the triangle ABC?

Answer 1

Answer:

The perimeter of the triangle ABC is 84 cm

Step-by-step explanation:

Given:

AD = 15cm

DC = 20 cm

Angle bisector theorem:

When vertical angle of a triangle is bisected, the bisector divides the base into two segments which have the ratio as the order of other two sides.

Apply angle bisector theorem in ΔABC, we have

$\frac{AB}{BC}=\frac{15}{20}$

$\frac{AB}{BC}=\frac{3}{4}$

$AB=3k\text{ and }BC=4k$

Apply pythagoras theorem in ΔABC, we have

$AC^2=AB^2+BC^2$

$35^2=(3k)^2+(4k)^2$

$1225=9k^2+16k^2$

$1225=25k^2$

$k^2=49$

$\implies\,k=7$

$AB=3(7)=21\,cm$

$BC=4(7)=28\,cm$

$\textbf{Perimeter of the triangle ABC}$

$=AB+BC+AC$

$=21+28+35$

$=84\,cm$