Please solve q 99?

enter image source here

2 Answers
Jun 17, 2018

#DP+EP+PF=5sqrt3# regardless where P is in the triangle.
This is called Viviani's theorem: https://en.wikipedia.org/wiki/Viviani%27s_theorem

Explanation:

enter image source here
ABC is the triangle. It's height #h=5sqrt3# (use Pythagoras on for instance ADC)

Remember that the area of a triangle is #(base*height)/2#.

Now all three triangles APC, BPC and APB have a base = 10 since all sides are equal and 10. Their total area is equal to the area of ABC

therefore #(side*CM)/2=(side*h)/2#
=#(side*DP)/2+(side*EP)/2*(side*FP)/2#

Therefore #DP+EP+PF=MC=5sqrt3# regardless where P is in the triangle.

Jun 17, 2018

Another one from Rahul's Book!

I'd say that given we're given choices, the result doesn't depend on our choice of #P#.

Let's pick #P=G,# the centroid/orthocenter. We get an altitude #5^2+h^2=10^2# or #h=5 sqrt{3}#. The medians meet #2/3# from each vertex, and the remaining piece of #1/3h# is perpendicular to the sides. We have three of these to add up, they add to #h#, choice (2).