Question #0994c

1 Answer
Nov 7, 2017

#r = 625/48" cm"#

Explanation:

Orient the triangle so that BC, is on the x axis, point B is #(-7,0)#, and point C is #(7,0)#; this makes the y axis the perpendicular bisector of chord BC at the origin #O = (0,0)#. Because the y axis forms a right triangle, #DeltaOAC#, with side AC as the hypotenuse, we can use the Pythagorean Theorem to find the coordinates of point A:

#AC^2= OC^2 + OA^2#

#25^2= 7^2+ OA^2#

#OA = sqrt(625 - 49)#

#OA = 24#

The coordinates of point #A = (0,24)#

Here is a graph of what I have described thus far:

www.desmos.com/calculator

Please observe that, the points A, B, and C are in black, the sides of #DeltaABC# are in green, and the perpendicular bisector of side BC is in purple.

The slope of the line AC is the slope from point #C= (7,0)# to point #A = (0,24)#:

#m = (24-0)/(0-7)#

#m = -24/7#

The slope, n, of its perpendicular bisector is:

#n = -1/m#

#n = -1/(-24/7)#

#n = 7/24#

The perpendicular bisector will go through the midpoint between A and C:

#((0+7)/2, (24+0)/2) = (3.5,12)#

The point-slope form of the equation of the perpendicular bisector is:

#y = 7/24(x - 3.5)+12#

Here is a graph of the triangle with the perpendicular bisector:

www.desmos.com/calculator

The center of the circle is the point where this line intercepts the y axis:

#y = 7/24(0 - 3.5)+12#

#y = 527/48#

The radius of the circle is the distance from the y intercept to point A

#r = 24 - 527/48#

#r = 625/48" cm"#

Here is a graph with the circle added.

www.desmos.com/calculator