Question

point a is a vertex of a polygon. point a is the image of a after the dilation. find the scale factor:

1. a(3,4) and a(9,12)
2. a(9,12) and a(6,8)
3. a(-2,-3) and a(-10,-15)
4. a(11,1) and a(22,2)
5. a(-4,-8) and a(-1,-2)
6. a(1,12) and a(0.75,9)

Explanation:

Step1: Recall scale - factor formula

For a dilation of a point \(A(x,y)\) to \(A'(x',y')\) with a scale - factor \(k\), we have \(x' = kx\) and \(y'=ky\). We can find \(k\) by dividing the coordinates of \(A'\) by the corresponding coordinates of \(A\) (assuming \(x
eq0\) and \(y
eq0\)). In general, \(k=\frac{x'}{x}=\frac{y'}{y}\).

Step2: Solve for problem 5

Given \(A(3,4)\) and \(A'(9,12)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{9}{3}=3\), for the \(y\) - coordinates: \(k = \frac{y'}{y}=\frac{12}{4}=3\). So the scale factor is \(3\).

Step3: Solve for problem 6

Given \(A(9,12)\) and \(A'(6,8)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{6}{9}=\frac{2}{3}\), for the \(y\) - coordinates: \(k=\frac{y'}{y}=\frac{8}{12}=\frac{2}{3}\). So the scale factor is \(\frac{2}{3}\).

Step4: Solve for problem 7

Given \(A(-2,-3)\) and \(A'(-10,-15)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{-10}{-2}=5\), for the \(y\) - coordinates: \(k=\frac{y'}{y}=\frac{-15}{-3}=5\). So the scale factor is \(5\).

Step5: Solve for problem 8

Given \(A(11,1)\) and \(A'(22,2)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{22}{11}=2\), for the \(y\) - coordinates: \(k=\frac{y'}{y}=\frac{2}{1}=2\). So the scale factor is \(2\).

Step6: Solve for problem 9

Given \(A(-4,-8)\) and \(A'(-1,-2)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{-1}{-4}=\frac{1}{4}\), for the \(y\) - coordinates: \(k=\frac{y'}{y}=\frac{-2}{-8}=\frac{1}{4}\). So the scale factor is \(\frac{1}{4}\).

Step7: Solve for problem 10

Given \(A(1,12)\) and \(A'(0.75,9)\), for the \(x\) - coordinates: \(k=\frac{x'}{x}=\frac{0.75}{1}=0.75=\frac{3}{4}\), for the \(y\) - coordinates: \(k=\frac{y'}{y}=\frac{9}{12}=\frac{3}{4}\). So the scale factor is \(\frac{3}{4}\).

Answer:

1. \(3\)
2. \(\frac{2}{3}\)
3. \(5\)
4. \(2\)
5. \(\frac{1}{4}\)
6. \(\frac{3}{4}\)