Question

Graph triangle JKL with vertices J(3,3), K(0,0) and L(0,-3) and it’s image after a dilation with scale factor k. List the new coordinates of the image below the graph. I need help with question 7.

Answer 1

Lets start by graphing the triangle. We will add the points first and then connect them.

The points are:

`\begin{gathered} J=(3,3) \\ K=(0,0) \\ L=(0,-3) \end{gathered}`

So we will get the following:

Now, the dilatation will be about point (-3, 6). It is easier to make the dilatation about the origin, so we can first translate the points (-3, 6) to the origin (0, 0), make the dilatation about the origin and then translate it back to (-3, 6).

Translate (-3, 6) to (0, 0) means to add 3 to the x coordinate and substract 6 from the y coordinate. We will apply this to all vertexes of the triangle:

`\begin{gathered} Center\colon(-3,6)\to(-3+3,6-6)\to(0,0) \\ J\colon(3,3)\to(3+3,3-6)\to(6,-3) \\ K\colon(0,0)\to(0+3,0-6)\to(3,-6) \\ L\colon(0,-3)\to(0+3,-3-6)\to(3,-9) \end{gathered}`

We get this:

Now, we apply the dilatation about (0, 0) (remeber that our center is now at the origin). The scale factor is 1/3, so we just multiply all the coordinates by it:

`\begin{gathered} Center\colon(0,0)\to(0\cdot(1)/(3),0\cdot(1)/(3))\to(0,0) \\ J\colon(6,-3)\to(6\cdot(1)/(3),-3\cdot(1)/(3))\to(2,-1) \\ K\colon(3,-6)\to(3\cdot(1)/(3),-6\cdot(1)/(3))\to(1,-2) \\ L\colon(3,-9)\to(3\cdot(1)/(3),-9\cdot(1)/(3))\to(1,-3) \end{gathered}`

We get:

And now we need to translate the center back to (-3, 6), which we do by substracting 3 from x and adding 6 to y (the contrary as we did before), we do this for all the vertexes too:

`\begin{gathered} Center\colon(0,0)\to(0-3,0+6)\to(-3,6) \\ J\colon(2,-1)\to(2-3,-1+6)\to(-1,5) \\ K\colon(1,-2)\to(1-3,-2+6)\to(-2,4) \\ L\colon(1,-3)\to(1-3,-3+6)\to(-2,3) \end{gathered}`

Drawing this, we have: