Geometry Problem: Finding the Sum of Perpendiculars in an Equilateral Triangle

[sankalpmittal]

Homework Statement

Let ABC be an equilateral triangle with side 10 cm and let P be a point inside the triangle at a distance of 2 cm from the side AB. Given --> AB = BC = CA = 10cm and PD = 2 cm. If PD, PE, PF are the perpendiculars to the three sides, find out the sum, PD + PE + PF.

Here is the image : http://postimage.org/image/1lt6hjgw4/

Homework Equations

I am not sure which equation is most relevant being a geometry question.

The Attempt at a Solution

I tried using similarity of triangles and Pythagoras theorem and trigonometry. I tried messing up all these things but failed to get the solution.

 

[ehild]

Try to express to area of the triangle with PE, PD and PF and with the side of the equilateral triangle.

ehild

 

[sankalpmittal]

Try to express to area of the triangle with PE, PD and PF and with the side of the equilateral triangle.

ehild

This is how I worked : http://postimage.org/image/1wf8ya2sk/
Am I correct ?

I found these equations :

x + 2*sqrt(3)/3 = r+y
x+y+r = 10
x-r-y= - 2*sqrt(3)/3

Am I Correct ?

 

[ehild]

I can not follow you. Why did you note by x both the upper and lower parts on the left-hand side of the triangle? The problem can be certainly solved by using similar triangles, but it would be complicated.

Why do you not try the area method? What are the areas of the coloured triangles in terms of PD, PE, PF? What is the area of the big triangle?


ehild

 

[sankalpmittal]

I can not follow you. Why did you note by x both the upper and lower parts on the left-hand side of the triangle? The problem can be certainly solved by using similar triangles, but it would be complicated.

Why do you not try the area method? What are the areas of the coloured triangles in terms of PD, PE, PF? What is the area of the big triangle?


ehild

OK , I get it .

Area of green Δ = 1/2 x 10 x PE = 5PE
Area of yellow Δ = 1/2 x 10 x PF = 5PF
Area of light blue Δ = 1/2 x 10 x PD = 5PD

Area of big Δ = sqrt(3)/4 x 100 = 25 sqrt(3)

5(PE + PF + PD) = 25 sqrt(3)
PE + PF + PD = 8.66025 cm approx.

Now , I can't understand why they gave me the length of PD. There was no need of it.

Thanks for the efforts.

 

[ehild]

PE + PF + PD = 8.66025 cm approx.

Now , I can't understand why they gave me the length of PD. There was no need of it.

To confuse the student. Really. Those evil teachers (including myself) do such things on purpose.

ehild

 

[sankalpmittal]

To confuse the student. Really. Those evil teachers (including myself) do such things on purpose.

ehild

Really ? Well that's strange and evil for sure

Anyways , thanks for your efforts !

 

[ehild]

This is a very good picture of me. Have we met before?

ehild

 

[sankalpmittal]

This is a very good picture of me. Have we met before?

ehild

No.

That is just a smiley which I think you liked.

Thanks again.