# A problem on geometry.

1. Oct 11, 2011

### sankalpmittal

1. The problem statement, all variables and given/known data

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/ [Broken]

2. Relevant equations

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

3. 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.

Last edited by a moderator: May 5, 2017
2. Oct 11, 2011

### ehild

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

ehild

3. Oct 11, 2011

### sankalpmittal

This is how I worked : http://postimage.org/image/1wf8ya2sk/ [Broken]
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 ?

Last edited by a moderator: May 5, 2017
4. Oct 11, 2011

### 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

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Last edited: Oct 12, 2011
5. Oct 12, 2011

### sankalpmittal

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.

6. Oct 12, 2011

### ehild

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

ehild

7. Oct 13, 2011

### sankalpmittal

Really ? Well that's strange and evil for sure

Anyways , thanks for your efforts !

8. Oct 13, 2011

### ehild

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

ehild

9. Oct 14, 2011

### sankalpmittal

No.:rofl:

That is just a smiley which I think you liked.

Thanks again.