find the missing endpoint if s is the midpoint of rt r(-9,4) and s(2,-1) ; find T
The Correct Answer and Explanation is:
To find the missing endpoint T(x,y)T(x, y)T(x,y) given:
- One endpoint R(−9,4)R(-9, 4)R(−9,4)
- Midpoint S(2,−1)S(2, -1)S(2,−1)
We use the midpoint formula:S=(x1+x22,y1+y22)S = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)S=(2×1+x2,2y1+y2)
Let R=(x1,y1)=(−9,4)R = (x_1, y_1) = (-9, 4)R=(x1,y1)=(−9,4), and let T=(x2,y2)T = (x_2, y_2)T=(x2,y2). We’re told the midpoint S=(2,−1)S = (2, -1)S=(2,−1). Plug the known values into the formula:
Step 1: Solve for x2x_2x2
−9+x22=2\frac{-9 + x_2}{2} = 22−9+x2=2
Multiply both sides by 2:−9+x2=4-9 + x_2 = 4−9+x2=4
Add 9 to both sides:x2=13x_2 = 13×2=13
Step 2: Solve for y2y_2y2
4+y22=−1\frac{4 + y_2}{2} = -124+y2=−1
Multiply both sides by 2:4+y2=−24 + y_2 = -24+y2=−2
Subtract 4 from both sides:y2=−6y_2 = -6y2=−6
Final Answer:
T=(13,−6)T = (13, -6)T=(13,−6)
Explanation:
This formula essentially means the midpoint lies exactly halfway between the two endpoints. If one endpoint and the midpoint are known, we can treat this formula like a system of equations to solve for the missing endpoint.
To find a missing endpoint of a line segment when the midpoint and one endpoint are given, we use the midpoint formula, which finds the average of the x-coordinates and y-coordinates of two endpoints:Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)Midpoint=(2×1+x2,2y1+y2)
In this problem, we are given:
- One endpoint R(−9,4)R(-9, 4)R(−9,4)
- The midpoint S(2,−1)S(2, -1)S(2,−1)
Let T(x,y)T(x, y)T(x,y) be the missing endpoint.
We apply the midpoint formula:
- For the x-coordinates: −9+x2=2\frac{-9 + x}{2} = 22−9+x=2. Solving gives x=13x = 13x=13.
- For the y-coordinates: 4+y2=−1\frac{4 + y}{2} = -124+y=−1. Solving gives y=−6y = -6y=−6.
So, the coordinates of the missing endpoint TTT are (13,−6)(13, -6)(13,−6).
This process is reliable and works for any situation involving midpoints and endpoints on the coordinate plane. It helps in geometry problems, computer graphics, engineering, and real-world spatial reasoning tasks.
