Author Topic: Trigonometry: Fitzpatrick 3U EX 4.1 (Read 1235 times) Tweet Share

annaconda · « **on:** July 14, 2017, 08:58:42 pm »

Hello!

I am stuck for this question:

P, A, B, C are four points in a plane such that the angles BPA and CPA are obtuse and on opposite sides of PA. PA = 8CM, BP = 10 CM, PC = 12 CM, AB = 14CM, and AC = 18CM. Calculate the length of BC and the area of the triangle ABC.

Thanks in advanced!!

RuiAce · « **Reply #1 on:** July 14, 2017, 09:35:30 pm »

$\text{Sketch a diagram carefully showing all information given and the length }BC\\ \text{It will look somewhat like a tetrahedron you can see through.}$
$\text{Applying the cosine rule in }\triangle BPA\\ \cos \angle BPA = \frac{100+64-196}{2\times 10\times 8} \implies\boxed{ \angle BPA = \cos^{-1}\left(-\frac15\right)}$
$\text{Applying the cosine rule in }\triangle CPA\\ \cos \angle CPA = \frac{144+64-324}{2\times 12\times 8}\implies \boxed{\angle CPA = \cos^{-1}\left(-\frac{29}{48}\right)}$
$\text{Since we know that }\angle BPC + \angle CPA + \angle APB=360^\circ\\\text{ as they are angles meeting at a point}\\ \boxed{\angle BPC = 360^\circ -\cos^{-1}\left(-\frac15\right) - \cos^{-1}\left(-\frac{29}{48}\right)}\\ \text{You can then use the cosine rule again to deduce your required answer. }\\\boxed{BC^2 = 100 + 144 - 2\times 10\times 12\times \cos \angle BPC}$
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$\text{To use }A=\frac12ab\sin C\\ \text{We require some angle. Suppose we want }\angle BAC$
$\text{Using the sine rule in the relevant triangles, we have}\\ \begin{align*}\angle BAP &=\sin^{-1}\left(\frac{10 \sin \angle BPA}{14}\right)\\ \angle CAP &= \sin^{-1}\left(\frac{12\sin \angle CPA}{18}\right) \end{align*}$
$\text{Of course, }\angle BAC\text{ is just the sum of the two above}\\ \text{and then }A = \frac{1}{2}\times 14\times 18 \times \sin \angle BAC$

annaconda · « **Reply #2 on:** July 14, 2017, 10:07:05 pm »

thank you!!!

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Author Topic: Trigonometry: Fitzpatrick 3U EX 4.1 (Read 1235 times) Tweet Share

annaconda

Trigonometry: Fitzpatrick 3U EX 4.1

RuiAce

Re: Trigonometry: Fitzpatrick 3U EX 4.1

annaconda

Re: Trigonometry: Fitzpatrick 3U EX 4.1

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