2 triangles

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Nov 7th 2009, 09:41 PM
thereddevils

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2 triangles

THe diagram shows triangle ABC where points A , B , C and P are coplanar . BC=BP=AP and angle BAC =angle ABP = angle PBC = pi/5 rad

Prove that triangle ABC and BPC are similar triangles . Hence deduce that

(1) BC^2=CP.CA

(2) $\cos (\frac{\pi}{5})=\frac{1}{4}(\sqrt{5}+1)$

I am only stucked with (2) , i am ok with the rest . THanks .
Nov 7th 2009, 11:20 PM
ukorov

if your diagram is right, then BC = BP is wrong.
Nov 9th 2009, 05:42 AM
thereddevils

Quote:

Originally Posted by ukorov

if your diagram is right, then BC = BP is wrong.

why is it so
Nov 9th 2009, 10:15 AM
ukorov

Quote:

Originally Posted by thereddevils

why is it so

oh alright not wrong actually, but could have been more detailed and more accurate. (Giggle)
Nov 10th 2009, 01:38 AM
Grandad

Hello thereddevils

Quote:

Originally Posted by thereddevils

THe diagram shows triangle ABC where points A , B , C and P are coplanar . BC=BP=AP and angle BAC =angle ABP = angle PBC = pi/5 rad

Prove that triangle ABC and BPC are similar triangles . Hence deduce that

(1) BC^2=CP.CA

(2) $\cos (\frac{\pi}{5})=\frac{1}{4}(\sqrt{5}+1)$

I am only stucked with (2) , i am ok with the rest . THanks .

(2) Suppose $BP=BC=AP = d$ . Then in $\triangle BPC$
$\angle BCP = \tfrac{2\pi}{5}$

$\Rightarrow PC = 2d\cos(\angle BCP) = 2d\cos(\tfrac{2\pi}{5})$

So, using the result in (1):
$d^2 = 2d\cos(\tfrac{2\pi}{5})\Big(d+2d\cos(\tfrac{2\pi}{ 5})\Big)$

$\Rightarrow 1 = 2\cos(\tfrac{2\pi}{5})+4\cos^2(\tfrac{2\pi}{5})$

Solving the quadratic, taking the positive root gives
$\cos(\tfrac{2\pi}{5})=\frac{\sqrt5-1}{4}$

$\Rightarrow 2\cos^2(\tfrac{\pi}{5})-1 = \frac{\sqrt5-1}{4}$

$\Rightarrow \cos^2(\tfrac{\pi}{5})= \frac{\sqrt5+3}{8}$
$=\frac{5+2\sqrt5+1}{16}$

$=\frac{(\sqrt5+1)^2}{16}$

$\Rightarrow \cos(\tfrac{\pi}{5}) = \frac{\sqrt5+1}{4}$

Grandad
Nov 10th 2009, 06:37 PM
ukorov

Quote:

Originally Posted by Grandad

Solving the quadratic, taking the positive root gives

$\cos(\tfrac{2\pi}{5})=\frac{\sqrt5-1}{4}$

$\Rightarrow 2\cos^2(\tfrac{\pi}{5}) -1 = \frac{\sqrt5-1}{4}$

why $\cos(\tfrac{2\pi}{5}) = 2\cos^2(\tfrac{\pi}{5})-1$ ??
Nov 10th 2009, 09:58 PM
Grandad

Hello ukorov

Quote:

Originally Posted by ukorov

why $\cos(\tfrac{2\pi}{5}) = 2\cos^2(\tfrac{\pi}{5})-1$ ??

Because $\cos2\theta = 2\cos^2\theta -1$ .

Grandad