Triangle A has an area of #12 # and two sides of lengths #3 # and #8 #. Triangle B is similar to triangle A and has a side of length #15 #. What are the maximum and minimum possible areas of triangle B?

1 Answer
Jun 30, 2018

Maximum possible area of triangle B is #300 # sq.unit
Minimum possible area of triangle B is
#36.99 # sq.unit

Explanation:

Area of triangle #A# is #a_A=12#

Included angle between sides #x=8 and z=3# is

#(x*z*sin Y)/2=a_A or (8*3*sin Y)/2=12 :. sin Y =1#

# :. /_Y= sin^-1(1)=90^0# Therefore, Included angle between

sides #x=8 and z=3# is #90^0#

Side #y=sqrt(8^2+3^2)= sqrt 73#. For maximum area in triangle

#B# Side #z_1=15# corresponds to lowest side #z=3#

Then #x_1=15/3*8=40 and y_1= 15/3*sqrt 73= 5 sqrt 73#

Maximum possible area will be #(x_1*z_1)/2=(40*15)/2=300 #

sq. unit. For minimum area in triangle #B# Side #y_1=15#

corresponds biggest side #y=sqrt 73#

Then #x_1=15/sqrt73*8=120/sqrt73# and

# z_1= 15/sqrt73*3= 45/ sqrt 73#. Minimum possible area will be

#(x_1*z_1)/2=1/2*(120/sqrt73* 45/ sqrt 73)= (60*45)/73#

#~~ 36.99 (2 dp)# sq.unit [Ans]