Beth+and+Maddie

=Beth K and Maddie S = =Second Quarter Project = =Hour 7 = == = =
 * Triangle ABC**

measure <A= 54 degrees measure <B= 71 degrees measure <C= 55.5 degrees

line segment AC= 7.3 cm line segment BC=6.3 cm line segment BA= 6.4 cm

We picked random points to make acute, scalene triangle ABC. To make sure our triangle was acute, we measured all the angles in Geometer's Sketch Pad to make sure they were less than 90 degrees. To make sure our triangle was scalene, we measured all the line segments in Geometer's Sketch Pad to make sure they were all different.

To calculate the midpoints D and E, we found the average x and y values of line segment AC and line segment BC.

x coordinate: (1+6)=7/2=3.5 y coordinate: (3+8)=11/2=5.5 midpoint D= (3.5,5.5)
 * To find point D on line segment AC...**

x coordinate: (7+6)=13/2=6.5 y coordinate: (2+8)=10/2=5 midpoint E= (6.5,5)
 * To find point E on line segment BC...**

We found the slope and perpendicular bisects of the line segments. For the slopes, we found the change in y over the change in x. For the perpendicular bisectors, we found the systems of equations of the line segments.


 * Slope of line segment AC...**

A (1,3) - C (6,8) = (-5,-5)=5/5=1


 * Perpendicular bisector of line segment AC...**

y= -1x+b 5.5= -1(3.5)+b 5.5= -3.5+b +3.5 +3.5 9=b

y= -1x+9


 * Slope of line segment BC...**

B (7,2) C (6,8) = (1,-6)= -6


 * Perpendicular bisector of line segment BC...**

y=1/6x+b 5=1/6(6.5)+b 5=13/12+b -13/12 -13/12 47/12=b

y=1/6x+47/12

We found the circumcenter by using the perpendicular bisectors of the line segments, to find where the two lines intersect. By using the distance formula, we found the distance to the circumcenter from each vertex.


 * Circumcenter...**

-1x+9=1/6x+47/12 -9 -9 -1x=1/6x-61/12 -1/6 -1/6 (-7/6)x=(-61/12) / (-7/6) (-7/6) x=61/14

y= -1(61/14)+9 y= -61/14+9 y=65/14

(61/14,65/14)=circumcenter [point F]


 * Distance Formula...**

A (1,3) -F (61/14,65/14) = (-47/14,-23/14)

(-47/14)^2 (-23/14)^2 2209/196 + 529/196 = square root of 1369/98 [which is about 3.737564415]