Final answer:
The two possible locations for point S that divides overline QR with endpoints Q (2, 12) and R (7, 6) into a 2:3 ratio are (4, 9.6) and (5, 8.4). These are achieved by applying the section formula with ratios of 2:3 and 3:2, respectively.
Step-by-step explanation:
To find the two possible locations of point S that divides overline QR into two parts with lengths in a ratio of 2:3, we can use the section formula for internal division. We have the endpoints Q (2, 12) and R (7, 6). Since the ratio is 2:3, we need to split the line segment into two parts where one is twice as long as the other. For the first part, let's consider S to be closer to Q. We divide the x-coordinates and y-coordinates using the ratio 2:3: x-coordinate of S = (2*7 + 3*2) / (2 + 3) = (14 + 6) / 5 = 20 / 5 = 4. y-coordinate of S = (2*6 + 3*12) / (2 + 3) = (12 + 36) / 5 = 48 / 5 = 9.6. Now, if S is closer to R, we invert the ratio and use 3:2: x-coordinate of S = (3*7 + 2*2) / (3 + 2) = (21 + 4) / 5 = 25 / 5 = 5. y-coordinate of S = (3*6 + 2*12) / (3 + 2) = (18 + 24) / 5 = 42 / 5 = 8.4.
Therefore, the two possible locations for point S are (4, 9.6) and (5, 8.4), which correspond to options a and d. To find the possible locations of point S, we need to divide the line segment QR into two parts with lengths in a ratio of 2:3. From the given coordinates of Q (2, 12) and R (7, 6), we can find the coordinates of S. First, we calculate the total length of QR using the distance formula: QR = sqrt((7-2)^2 + (6-12)^2) = sqrt(25 + 36) = sqrt(61). Next, we calculate the distance from Q to S (QS) using the ratio 2:3, where QS = (2/5) * QR. Finally, we use the midpoint formula to find the coordinates of S: (x,y) = (2 + ((2/5) * (7-2)), 12 + ((2/5) * (6-12)) = (5, 9.6) or (4, 8.4).To find the possible locations of point S, we need to divide the line segment QR into two parts with lengths in a ratio of 2:3. From the given coordinates of Q (2, 12) and R (7, 6), we can find the coordinates of S.