Question

I need to know what the scale factor is so

Answer 1

Since triangle STU is a reduction of the triangle PQR, their corresponding sides are related by the equation:

`S_{\text{PQR}}=kS_{\text{STU}}`

Where K is a constant, and that constant is the same whichever pair of corresponding sides we choose.

Let's take the PQR side, its length is given by:

`\begin{gathered} P=(-8,4),R=(-4,-2) \\ \Rightarrow d_(PR)=\sqrt[]{(-8+4)^2+(4+2)^2}=\sqrt[]{16+36}=\sqrt[]{52} \end{gathered}`

On the other hand, the length of the SU side is:

`\begin{gathered} S=(-2,1),U\approx(-1,-1) \\ \Rightarrow d_(SU)\approx\sqrt[]{(-2+1)^2+(1+1)^2}=\sqrt[]{1+4}=\sqrt[]{5} \end{gathered}`

Then, we need to find the value of c in the equation

`\begin{gathered} d_(PR)=c\cdot d_(SU) \\ \Rightarrow c=(d_(PR))/(d_(SU))\approx\sqrt[]{(52)/(5)}=\sqrt[]{10.4}=3.22490\ldots \\ \Rightarrow c\approx3.225 \end{gathered}`

Then, as STU is the reduction of PQR, the answer is 1/c, which is:

`\begin{gathered} (1)/(c)\approx\frac{1}{\sqrt[]{10.4}}=0.310087 \\ \Rightarrow(1)/(c)\approx0.31 \end{gathered}`

The reduction factor is approximately 0.31. For a better result, we need to know with more certainty the coordinates of U