Triangle PQR is a 30-60-90 triangle with right angle Q and segment PQ as the longer leg. Find the possible coordinates of R if (2,6) and Q(2,-6).
To find the possible coordinates of point R in triangle PQR, we can use the properties of a 30-60-90 triangle. In this type of triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg.
First, let's find the length of segment PQ, which is the longer leg. The coordinates of point Q are (2, -6), and the coordinates of point P are (2, 6). Using the distance formula, we can find the length of segment PQ:
Length of PQ = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 2)^2 + (-6 - 6)^2]
= √[0^2 + (-12)^2]
= √(0 + 144)
= √144
= 12
Now, since point R is on the hypotenuse, which is twice the length of the shorter leg, the length of segment QR would be 2 times the length of segment PQ:
Length of QR = 2 * 12
= 24
Since point Q is at (2, -6), we can find two possible points for R by moving 24 units in both the positive and negative y-direction from point Q.
Possible coordinates for point R: (2, -6 + 24) = (2, 18) and (2, -6 - 24) = (2, -30).
Therefore, the two possible coordinates for point R in triangle PQR are (2, 18) and (2, -30).