1. ## HELP!! please and thank you

A crew is surveying Mr. Gournic's property. From his front door, 10 m away on a bearing S 48deg. E is his mailbox. Seven m from the mailbox on a bearing of S 42 deg. W is a telephone pole. a) How far is it from the door directly to the pole? (exact value) b) What is the bearing from the pole to the door?

2. Originally Posted by Luckegrl_16
A crew is surveying Mr. Gournic's property. From his front door, 10 m away on a bearing S 48deg. E is his mailbox. Seven m from the mailbox on a bearing of S 42 deg. W is a telephone pole. a) How far is it from the door directly to the pole? (exact value) b) What is the bearing from the pole to the door?
In the diagram, let H be Mr. Gournic's House, L be his letterbox, and P be the telephone pole.

By alternate angles on parallel lines, we can deduce various other angles in the diagram.
After filling in several angles, we note that $\displaystyle \angle HLP = 42 + 48 = 90^{\circ}$, so the paths actually form a right angled triangle $\displaystyle \Delta HLP$.

Then, by Pythagoras' Theorem,

$\displaystyle HP =\sqrt{10^2+7^2}=\sqrt{149}$

So the distance from his door to the pole is $\displaystyle \sqrt{149}$ m.

Next, to find the beaing of H from P, note that

$\displaystyle \tan{(\angle LHP)} = \frac{7}{10}$, so

$\displaystyle \angle LHP = \arctan (\frac{7}{10})$

Now, note that by alternate angles on parallel lines, $\displaystyle x^{\circ} =48- \angle LHP = (48-\arctan (\frac{7}{10}))^{\circ}$

So $\displaystyle x= 13^{\circ}$, approximately.

Therefore the bearing to H from P is $\displaystyle 360 - 13 = 347^{\circ}$, or, if you prefer, $\displaystyle N \ 13^{\circ} W$

3. ## *t*h*a*n*k* *y*o*u*

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