Question

randy draws triangle abc on the coordinate plane with vertices a(7, -4), b(10, 3), and c(6, 1). he then translates the figure so the coordinates of the image are a(5, 1), b(8, 8), and c(4, 6). what rule did he use to draw the image?

t_{-5,2}(x,y)

t_{-2,5}(x,y)

t_{2,-5}(x,y)

t_{5,-2}(x,y)

Explanation:

Step1: Calculate the change in \(x\) - coordinate

For point \(A(7,-4)\) and \(A'(5,1)\):
The change in \(x\) - coordinate is \(5 - 7=-2\).
For point \(B(10,3)\) and \(B'(8,8)\):
The change in \(x\) - coordinate is \(8 - 10=-2\).
For point \(C(6,1)\) and \(C'(4,6)\):
The change in \(x\) - coordinate is \(4 - 6=-2\).

Step2: Calculate the change in \(y\) - coordinate

For point \(A(7,-4)\) and \(A'(5,1)\):
The change in \(y\) - coordinate is \(1-(-4)=5\).
For point \(B(10,3)\) and \(B'(8,8)\):
The change in \(y\) - coordinate is \(8 - 3=5\).
For point \(C(6,1)\) and \(C'(4,6)\):
The change in \(y\) - coordinate is \(6 - 1=5\).

The translation rule is \((x,y)\to(x - 2,y + 5)\), which is \(T_{-2,5}(x,y)\)

Answer:

\(T_{-2,5}(x,y)\) (the second option)