We know that â–ł ABC is and that A(3,2), and B(7,2). We want to find all possible coordinates of point C. First, let's plot point A and B on the coordinate grid.
Looking at the graph, we see that the distance between A and B is 4 units. Recall that a with at (x,y) and r is the set of all points that are in distance r from (x,y). Therefore, if we draw two circles with radii of 4 units, one with center at A, and the other one with center at B, the intersections of these circles will give us candidates for point C.
We have two candidates for point C: point C_1 above AB, and point C_2 below AB. Both candidates for point C have 5 as their x-coordinate. We will first find the y-coordinate of the candidate for C above AB.
First Candidate
Let's draw C_1C_2. This segment intersects AB at a right angle, dividing AB into two equal parts, each 2 units long. Finally, let's plot AC_1. Since C_1 lies on a circle with center at A and radius 4, we have AC_1 = 4. This added segments will create a with AC_1 as its .
Let's denote the other of this right triangle by h. We can use the to write an equation for h.
h^2 + 2^2 = 4^2
Let's solve it!
h^2 + 2^2 = 4^2
h^2 + 4 = 16
h^2 = 12
h = sqrt(12)
h = sqrt(4 * 3)
h = sqrt(4) * sqrt(3)
h = 2* sqrt(3)
h = 2sqrt(3)
We see that point C_1 is 2sqrt(3) units above AB.
Since both points A and B have a y-coordinate of 2, the y-coordinate of point C_1 is 2 + 2sqrt(3). Recall that both candidates for point C have 5 as their x-coordinate. This gives us the following coordinates for C_1.
C_1 = C_1( 5, 2 + 2sqrt(3))
Second Candidate
To find the y-coordinate of C_2, note that our diagram with two circles and their intersection is across AB.
Therefore, point C_2 is the same number of units below AB as C_1 is above this segment.
This time we should subtract 2sqrt(3) from 2, the common y-coordinate of A and B. In consequence, the y-coordinate of C_2 is 2 - 2sqrt(3). This gives us the following coordinates for C_2.
C_2 = C_2( 5, 2 - 2sqrt(3))
Conclusion
We have two points that are candidates for point C.
(5,2+sqrt(3)) and (5,2-sqrt(3))
