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Question 891984: Hi there.
I have this question here and I'm really struggling with it.
I would really appreciate if someone could help me, to sort it out.
Question.
Karen and her friends are playing rounders in the park. Jennifer is standing at point J, which is 18 metres due north of point C, where Claire is standing. Annette is standing at point A, 52 degrees east of south of Jennifer and 12.4 metres from where Jennifer is standing. (You may assume that all distances are flat and in a straight line.)
A. Draw a diagram showing the points J,C and A, marking in the angle and the lengths that you are given. Include the line CA in your diagram.
B. Find the distance between Claire an Annette, that is, the length of the line CA.
C. Karen moves to position K between Jennifer and Claire (on the line JC), due west of Annette (A). Add a line to your diagram that shows the shortest distance from Annette's position to the point where Karen is standing. What angles does KA make with the line JC?
D. Find which person is nearest to Karen.
Well I have to use trigonometry, not scale drawings, to find my answers.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your diagram is attached.
my comments are below the diagram.
m is the distance between point J and K.
n is the distance between point K and S.
p is the distance between point A and J.
q is the distance between point A and C.
o is the distance between point A and K.
Your questions will be answered below:
the diagram is shown above.
that's your answer to question A.
the rest of the questions couldn't be answered in turn because i needed to find other pieces of information before i could find the distance between point C and A.
in particular, the line AK needed to be drawn.
to find the distance between claire and annette, I needed to find the length of m and n and o.
m is the distance between points J and K.
n is the distance between points K and C.
o is the distance between point K and A.
the line AK formed 2 right triangles.
the first right triangle is JAK.
the second right triangle is CAK
the reason line AK is perpendicular to line JC is because line JC is due north and line AK is due west. this forms the right angle at K.
triangle JAK was used first.
sin(J) = opp/hyp = o/p
J = 52 degrees
p = 12.4 meters.
equation becomes sin(52) = o/12.4
multiply both sides of this equation by 12.4 to get:
12.4 * sin(52) = o
solve for o to get o = 9.771333345 meters.
that's the distance between points K and A.
cos(J) = adj/hyp = m/p
J = 52 degrees
p = 12.4 meters.
equation becomes cos(52) = m/12.4
multiply both sides of this equation by 12.4 to get:
12.4 * cos(52) = m
solve for m to get m = 7.634202294
that's the distance between points J and K
since m + n = 18, you can solve for m by subtracting m from 18.
when you do that, you get n = 18 - m = 10.36579771 meters.
that's the distance between K and C.
you have everything except q which is the distance between point A and C.
to find that, you need to find the measure of angle C.
the right triangle CAK was used for that.
since you now have o and n, you can solve for angle C and, once you have angle C, you can solve for q.
tan(C) = o/n
o = 9.771333345 meters.
n = 10.36579771 meters.
you get tan(C) = .9426513638
angle C = arctan(.9426513638) = 43.30907343 degrees.
now that you have angle C, you can find q.
sin(C) = opp/hyp = o/q
C = 43.3090743 degrees.
o = 9.771333345 meters.
equation becomes sin(43.3090743) = 9.771333345/q
multiply both sides of this equation by q to get:
q * sin(43.3090743) = 9.771333345
divide both sides of this equation by sin(43.3090743) to get:
q = 9.771333345 / sin(43.3090743) = 14.2453051
that's the distance between points A and C.
the distance between points A and C is 14.2453051 meters.
now i'll try to answer your questions.
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A. Draw a diagram showing the points J,C and A, marking in the angle and the lengths that you are given. Include the line CA in your diagram.
the diagram should answer your questions.
the line CA is in the diagram.
also the line AK is in the diagram.
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B. Find the distance between Claire an Annette, that is, the length of the line CA.
the distance between claire and annette is the distance between point C and A which is the length of line q which is equal to 14.2453051 meters.
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C. Karen moves to position K between Jennifer and Claire (on the line JC), due west of Annette (A). Add a line to your diagram that shows the shortest distance from Annette's position to the point where Karen is standing. What angles does KA make with the line JC?
the line KA makes a 90 degree angle with the line JC.
this is because the line KA is due west and the line JC is due north.
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D. Find which person is nearest to Karen.
karen is at point K.
jennifer is at point J.
annette is at pont A.
claire is at point C.
the distance between K and J is equal to m which is equal to 7.634202294 meters.
the distance between K and A is equal to o which is equal to 9.771333345 meters.
the distance between K and C is equal to n which is equal to 10.36579771
the person that is nearest to karen is jennifer.
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