The midpoint of ST is W(-3, 1). Point S is located at (-7,-9). What are the coordinates of point T?
Answer (2)
Use the midpoint formula to set up equations for the x and y coordinates of point T.
Solve the equation 2 − 7 + x = − 3 for x, which gives x = 1 .
Solve the equation 2 − 9 + y = 1 for y, which gives y = 11 .
State the coordinates of point T as ( 1 , 11 ) .
Explanation
Problem Analysis We are given that the midpoint of segment ST is W(-3, 1) and the coordinates of point S are (-7, -9). Our goal is to find the coordinates of point T. Let's denote the coordinates of point T as (x, y).
Midpoint Formula The midpoint formula states that the coordinates of the midpoint W of a line segment with endpoints S(x_1, y_1) and T(x_2, y_2) are given by:
W = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
Applying the Midpoint Formula In our case, W(-3, 1) is the midpoint of ST, and S is located at (-7, -9). Therefore, we have:
− 3 = 2 − 7 + x
1 = 2 − 9 + y
Solving for x Now, let's solve for x:
− 3 = 2 − 7 + x
Multiply both sides by 2:
− 6 = − 7 + x
Add 7 to both sides:
x = − 6 + 7
x = 1
Solving for y Next, let's solve for y:
1 = 2 − 9 + y
Multiply both sides by 2:
2 = − 9 + y
Add 9 to both sides:
y = 2 + 9
y = 11
Final Answer Therefore, the coordinates of point T are (1, 11).
Examples
In urban planning, understanding midpoints is crucial when designing transportation routes. If we consider two residential areas, S and T, the location of a new bus stop, W, is ideally placed at the midpoint to minimize the distance residents from both areas need to walk. Knowing the coordinates of S and the desired midpoint W, we can calculate the optimal location T for the second residential area to ensure balanced accessibility.
By using the midpoint formula, we determined that point T's coordinates are (1, 11) based on the given midpoint and point S. We calculated the x-coordinate using the equation derived from the formula, followed by the y-coordinate. The calculations led us to find the exact location for point T on the coordinate plane.
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