b Find the distance between P and Q, and consider the cases from Part A.
A
a (-1,5), (-10,8), (8,2)
B
b (-10,8). See solution.
Practice makes perfect
a We know the coordinates of points P and Q, so we can plot them and draw the segment PQ on the coordinate plane first. We can place the point R in the middle of the segment PQ. Let's do it by drawing a Perpendicular Bisector of the segment.
As we can see, (-1,5) are the coordinates of point R in this case. However, we can also make the points P or Q the midpoints of segments RQ and PR respectively. To do it, we can draw the line PQ first.
Now we take a compass, open it to length of PQ, and draw an arc that intersects the line by putting the compass point on point P or Q. If we put it on point P, we get the following graph and we can
identify the coordinates of R.
As we can see, when P is the midpoint of RQ, then (-10, 8) are the coordinates of point R. Let's do the same on the other side of the segment.
We can see that in this case (8,2) are the coordinates of point R. We summarize these cases in the table below.
Case
Coordinates of R
R is the midpoint
(-1,5)
P is the midpoint
(-10,8)
Q is the midpoint
(8,2)
b To see what how the length of RQ can change the answer from Part A, let's calculate the distance between the points P and Q. We can do it with the Distance Formula.
The distance between P and Q is sqrt(40). Notice that we can rewrite the distance between R and Q as an expression with sqrt(40).
sqrt(160) = sqrt(4 * 40) = 2 * sqrt(40)
Therefore, the distance between R and Q is twice the distance between P and Q. Let's analyze each case from Part A to determine which of them is still possible.
Case 1
The first case is when point R is the midpoint of segment PQ.
It divides it into two equal segments, thus RQ is half of the length of PQ. Since PQ is sqrt(40), then RQ would be sqrt(40)2. However, we know that RQ= 2 * sqrt(40), so this case is not possible!
Case 2
The second case is when P is the midpoint of the segment RQ.
We know that the length of PQ is sqrt(40). It is half of the segment RQ, so in this case RQ=2 * sqrt(40). This is the length we wanted! Therefore, this case is possible.
Case 3
The last case is when Q is the midpoint of PR.
We know that the length of PQ is sqrt(40), and it is the same as the length of the segment RQ. Thus RQ= sqrt(40), which in this case is not what we want.
Conclusion
As we can see, only one case satisfies the given condition. Therefore, (-10,8) are the only possible coordinates of R when RQ=sqrt(160).
