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Math Help - solving triangles- please check my answers!

  1. #1
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    Question solving triangles- please check my answers!

    1. How many solutions exist for the triangle where a=3.8, b=14.4, and A=157 degrees?
    Answer: No triangles

    2. Use the given measurements to solve triangle ABC: a=12, b=20, c=14.
    Answer: A= 36.2 degrees, B=100.3 degrees, and C=43.5 degrees.

    3. Find the value of x in the triangle with an area of 81.7 square units, a=17.5, and b=c=x.
    Answer: I'm having a lot of trouble with this problem. If b=c then angle B also = angle C. So I used the equation to find the semi-perimeter: (a+b+c)/2. Then i simplified that to be S=8.75+x. Then I plugged that value for S into Heron's area formula: square root [S(S-a)(S-b)(S-c)]. I get a very odd decimal for x that must be wrong. HELP!!!

    4. Find c if A=23 degrees, a=9, and b=12.
    Answer: I found that this is an example of the ambiguous case and that there will be 2 triangles. So I eventually got that c1=18.73 and c2=3.36.

    5. A ship travels due west for 83 miles. It then travels in a northern direction for 56 miles and ends up 115 miles from its original position. How many degrees did it turn when it changed direction?
    Answer: A= 110.1 degrees

    6. Find the area of triangle ABC if A=22 degrees, a=7.7, and b=7.7
    Answer:I found this to be another example of the ambiguous case (SSA). There will be two triangles and, therefore, two areas. I found the area of triangle 1 to be 20.59 square units. However, I am at a loss as to how to find the area of triangle 2. HELP!!!

    7. A plane travels 110 miles at a heading of North 40 degrees West (N40W). It then changes direction and travels 105 miles at a heading of N 66 degrees West (N66W). How far is the plane from its original position?
    Answer: What I don't understand about problems involving bearing is how to find the angle within the triangle. If the bearing is N40W, do I subtract that angle from 90 to get 50? Or do I subtract from 180? Same thing with N66W. If I could understand how to draw the diagram, then I would be able to solve the problem. HELP!!!

    8. IF the area of a triangle is 8.70 square units and A=56 degrees and b=7, find the length of c to the nearest integer.
    Answer: Using the equation Area=1/2bcsinA, I found c to be 3.

    9. Find the third side of the triangle if A=55 degrees, a=9.4, and b=9.4
    Answer: If a=b then A must also = B. Then C must = 70 degrees. Then, using the law of sines, i found c to be 10.78

    10. Two coast guard stations located 75 miles apart on a north to south line each receive a radio signal from a ship at sea. From the northernmost station, the ship's bearing is South 73 degrees East (S73E). From the other station, the ship's bearing is North 23.4 degrees East (N23.4E). How far is the ship from the southernmost station?
    Answer: Once again, this problem deals with bearings and I am at a loss. If I could just get the diagram drawn correctly, I think I could figure out the rest. HELP!!!

    11. Find the third side of the triangle with A=34 degrees, b=5, and c=8.
    Answer: Using the law of cosines, I found that a=4.76

    12. An airplane left an airport and flew east for 89 miles. Then it turned northward to North 17 degrees East (N17E). When it was 161 miles from the airport, there was an an engine problem and it turned to take the shortest route back to the airport. Find the angle through which the airplane turned.
    Answer: Again, this problem deals with bearing. I really need to understand how to draw the diagram. HELP!!!

    13. Find the length of the diagonal of the parallelogram with a base of 9.2 units. The right side of the parallelogram is 7.6 units. The entire bottom left angle is 65 degrees.
    Answer: So the diagonal is splitting the parallelogram into 2 congruent triangles, right? So, I believe that means it is bisection the angle that is 65 degrees and that each triangle is actually comprised of an angle of 32.5 degrees. You really only need to solve one triangle, though. The thing is, from what I can tell, that triangle is an example of the ambiguous case. So, to determine the number of triangles, I used the formula h=bsinA. I found h to be 4.94. Then, that means h<a<b, so there are 2 triangles??? I am really confused. So will there be 2 values for the diagonal? HELP!!!

    I really need help with the problems involving bearing and the ambiguous case!!!
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  2. #2
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    Quote Originally Posted by iheartthemusic29 View Post
    1. How many solutions exist for the triangle where a=3.8, b=14.4, and A=157 degrees?
    Answer: No triangles

    2. Use the given measurements to solve triangle ABC: a=12, b=20, c=14.
    Answer: A= 36.2 degrees, B=100.3 degrees, and C=43.5 degrees.

    3. Find the value of x in the triangle with an area of 81.7 square units, a=17.5, and b=c=x.
    Answer: I'm having a lot of trouble with this problem. If b=c then angle B also = angle C. So I used the equation to find the semi-perimeter: (a+b+c)/2. Then i simplified that to be S=8.75+x. Then I plugged that value for S into Heron's area formula: square root [S(S-a)(S-b)(S-c)]. I get a very odd decimal for x that must be wrong. HELP!!!

    ...
    #1 and #2 are OK.

    to #3. Since b = c the triangle is isosceles with base a. Calculate the height of the triangle perpendicular to the base:

    A = \frac12 \cdot a \cdot h~\implies~81.7 = \frac12 \cdot 17.5 \cdot h~\implies ~h = \frac{1634}{175} \approx 9.337...

    The triangle is divided into 2 right trinagles by the height. Therefore

    \left(\frac12 \cdot a\right)^2+h^2 = c^2~\implies~ c \approx 12,796..
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  3. #3
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    Quote Originally Posted by iheartthemusic29 View Post
    ...

    13. Find the length of the diagonal of the parallelogram (normally there are 2 different diagonals!) with a base of 9.2 units. The right side of the parallelogram is 7.6 units. The entire bottom left angle is 65 degrees.
    Answer: So the diagonal is splitting the parallelogram into 2 congruent triangles, right? right So, I believe that means it is bisection the angle that is 65 degrees and that each triangle is actually comprised of an angle of 32.5 degrees. (That is an heresy and you know what will happen to you...) HELP!!!
    The angles at the base are
    \alpha = 65^\circ and
    \beta = 180^\circ - \alpha = 115^\circ

    Now use Cosine rule twice:

    d_1 = \sqrt{9.2^2 + 7.6^2 - 2 \cdot 9.2 \cdot 7.6 \cdot \cos(65^\circ)} \approx 9.1269...

    d_2 = \sqrt{9.2^2 + 7.6^2 - 2 \cdot 9.2 \cdot 7.6 \cdot \cos(115^\circ)} \approx  14.195...
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  4. #4
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    Quote Originally Posted by iheartthemusic29 View Post
    ...

    4. Find c if A=23 degrees, a=9, and b=12.
    Answer: I found that this is an example of the ambiguous case and that there will be 2 triangles. So I eventually got that c1=18.73 and c2=3.36.

    5. A ship travels due west for 83 miles. It then travels in a northern direction for 56 miles and ends up 115 miles from its original position. How many degrees did it turn when it changed direction?
    Answer: A= 110.1 degrees

    6. Find the area of triangle ABC if A=22 degrees, a=7.7, and b=7.7
    Answer:I found this to be another example of the ambiguous case (SSA). There will be two triangles and, therefore, two areas. I found the area of triangle 1 to be 20.59 square units. However, I am at a loss as to how to find the area of triangle 2. HELP!!!

    ...
    #4 and #5 are OK.

    to #6: The triangle ABC is an isosceles triangle. Since a = b the angles A and B must be equal too. Therefore the angle C = 136.

    Now use the formula to calculate the area F (I have to use another letter than A or a):

    F = \frac12 \cdot a \cdot b \cdot \sin(C) . Plug in the values of a, b and C. You'll get

    F  \approx 20.593..... . So your answer is OK. But there isn't a second different triangle.
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