Okay, I worked it out. Although the calculations were tedious, it
works and you will be able to obtain value for all the question.
(Assuming that the "all sides of property" means the distances
travelled for each direction. In this case they ask for only the
direction he travelled after the old oak tree because others are
already given?) My method is tedious because I focused on solving
rather than convenience. So I will write the method and will let you
calculate because I prefer not to work with a calculator.
1 : I believe you could have at least drawn a diagram.
2 : (Angles) From the starting point: 80 (56+32) (180-22-56) and
(22+68). The hint is to utilize the parallel lines (which in this case
is used to refer the angle of direction headed by the person) The
angle of 32 degrees, of 22 degrees, and of 56 degrees are able to
equate with other (respective) angles.
(Lengths)
i. Drawing additional lines and EXTRA LABELS:
For this question the graph is cruicial but I assume you have that.
Suppose: from the starting point, label (a), (b) - first turn, (c) -
the old oak tree, (d) - reaching Mulberry Lane again. First draw
vertical line down from (b) and horizontal line to the right of (c)
until crossing the path (a)(b) - let the the distance until reaching
the path (a)(b) be called G and the intersection be called (e). and
the intersection of the vertical and horizontal be called (f). You
will then see smaller triangle (b)(f)(e). Let the hypothenus be now
called H, then the rest of length is 320-H. Next I want to connect (a)
and (c), and let the distance be called K. Lastly, label the angle
between G and K be "thetha".
ii. Finding K:
First, find the length of D. According to the pythagorean theorem (for
general triangle, not restricting to the right triangle) G can be
defined as:
G^2 = 280^2 + H^2 - 2(280)(H)cos(56+32)
therefore K can be expressed with G and 320-H by, again, the general
pythagorean theorem
K^2 = G^2 + (320-H)^2 -2(G)(320-H)cos(180-(90-32))
The angle for above equation can be simply found by using the smaller
triangle I earlier formed. The form is simple but contains two
variables. So you will need at least one more equation to express K
but in a different shape. Observer that you see another triangle
(a)(b)(c) K is the opposite of the nagle (56+32) Thus the K is written
as
K^2 = 280^2 + 320^2 - 2(280)(320)cos(56+32)
By solving the system of equations, you should be able to obtain both K and H.
Now, to get to the two sides you need (I assume) to find out, "theta"
because I will be using trig functions. Once again using the
pythagorean theorem, "theta" is easily defined:
(320-H)^2 = G^2 + K^2 - 2(G)(K)cos("theta")
320-H is known because H is known by previous system, G is known
because H is known by the first equation of G, K is known because the
system of equations allowed numerical value. Therefore there is really
only one variable, you can find it because there is exactly one
equation. After finding "theta", you will see the angle (d)(c)(a) is
102-(90-56)-"theta" and let it call T. Then two unknown sides are
respectively,
(c)(d) = K*cos(T)
(d)(a) = K*sin(T) - forgive me if I confused sine and cosine
By the way, if you misunderstood, I mean product when I put integers
in parentheses. I usually don't write Asterisk for product, so
(320)cos(56+32) means 320 TIMES cosine of 88.
3. For the area of property, I suggest you use sine's - (area) =
(1/2)(adjacent side 1)(adjacent side 2)sin(Angle). You could possibly
use subtraction of triangles from rectangle, but you should decide. I
re-constructed three individual triangles, so you should be able to
calculate area of each triangle to find the total area of property.
4. Tax depends on the answer you give in 3.
Good luck |