Can someone help me on this math problem?

Question

This is a problem that I need help with. I know how to do it but the problem is that I have to solve this problem without a calculator.

There is a boy who is standing at the center of a circular field that had a radius of 135 feet. He walks due north halfway to the circle. He then turns and walks due east half way to the circle. He turns again and walks due south half way to the circle. Finally he turns and walks due west half way to the circle. When he stops, how many feet is the boy from the center of the circle? Round the answer to the nearest foot.

Thanks for everyone who posted but no one has told me how to solve the problem with absoutely

NO USE of the calculator.


There is a boy who is standing at the center of a circular field that had a radius of 135 feet. He walks due north halfway to the circle. He then turns and walks due east half way to the circle. He turns again and walks due south half way to the circle. Finally he turns and walks due west half way to the circle. When he stops, how many feet is the boy from the center of the circle? Round the answer to the nearest foot.

Answer 1

[Revised: I initially made a mistake in the
midpoint calculation of the last two points,
which has since been corrected.]

Refer to the drawing at:
http://aycu06.webshots.com/image/32645/2002875752340123485_rs.jpg

If the starting point is the origin (0,0), then
the equation for the circle is:

x^2 + y^2 = 135^2

For the first "leg", he travels north 135/2 feet,
arriving at point:

location = (0, 135/2)
location = (0, 67.5)

For the second "leg", he travels east halfway to
the circle. Since his "y" coordinate is 67.5, we
need to find the point where the line y=67.5
intersects the circle.

x^2 + y^2 = 135^2
x^2 = 135^2 - y^2
x^2 = 135^2 - 67.5^2
x^2 = 135^2 - 67.5^2
x^2 = 13668.75
x = 116.91343

The point halfway to this point on the circle
is:

location = (116.91343/2, 67.5)
location = (58.4567, 67.5)

For the third leg, he travels south halfway to
the circle. Since his "x" coordinate is 58.4567,
we need to find the point where the line
x=58.4567 intersects the circle. Note that it
falls below the x-axis, so we want the negative
value of y:

x^2 + y^2 = 135^2
y^2 = 135^2 - x^2
y^2 = 135^2 - (58.4567)^2
y^2 = 18225-3417 = 14807
y = -121.687

To calculate the midpoint, find the distance
between (58.4567, 67.5) and (58.4567, -121.687):

d = 67.5 - (-121.687)
d = 67.5 + 121.687
d = 189.19

d/2 = 94.6

So the new point is:

location = (58.46, 67.5 - 94.6)
location = (58.46, -27.1)


For the final leg, he travels west halfway to the
circle. Since the "y" coordinate is -27.1 we
need to find the point where the line y=-27.1
intersects the circle. Note that it crosses the
y-axis, so we need to use the negative value:

x^2 + y^2 = 135^2
x^2 = 135^2 - (-27.1)^2
x^2 = 135^2 - 734.1
x^2 = 18225 - 734.1
x^2 = 17490.9
x = -132.25

So the final position will be halfway to the
point (-132.25,-27.1) from the point (58.46,
-27.1):

d = 58.46 - (-132.25)
d = 58.46 + 132.25
d = 190.7

d/2 = 95.4

Therefore, the final point is:

location = (58.5 - 95.4, -27.1)
location = (-36.9, -27.1)

To calculate the distance, use the Pythagorean
theorem:

D = sqrt(x^2 + y^2)
D = sqrt((-36.9)^2 + (-27.1)^2)
D = sqrt(1361.6 + 734.4)
D = sqrt(2096.0)
D = 45.78

Rounded to the nearest foot:

D = 46 feet

Answer 2

Look at the sketch:
http://s236.photobucket.com/albums/ff177/jsardi56/?action=view¤t=Circlepath.jpg

Radius = 135
a = 135/2 = 67.5
b = √(135^2 - 67.5^2) = 116.9134
c = d = 116.9134/2 = 58.45671
e = √(135^2 - 58.45672^2) = 121.68735
CD = 67.5 + 121.68735 = 189.18735
f = 189.18735/2 = 94.593675
g = 94.593675 - 67.5 = 27.09375
FH = √(135^2 - 27.093675^2) = 132.25329
h = 95.355 - 58.45671 = 36.89829
j = √(36.89829^2 + 27.093675^2) = 45.777189feet