Question

dilate triangle lmn with a scale factor of 1/3 using point p as the center of dilation. to dilate triangle lmn, you can dilate the vertices and then connect the resulting points. start with vertex l. plot point l, the image of point l after this dilation.

Explanation:

Step1: Recall dilation formula

If the center of dilation is \(P(x_p,y_p)\) and the point to be dilated is \(L(x_l,y_l)\) with a scale - factor \(k\), the formula for the image \(L'(x_{l'},y_{l'})\) is \(x_{l'}=x_p + k(x_l - x_p)\) and \(y_{l'}=y_p + k(y_l - y_p)\). First, assume the grid has a unit - length of 1 and find the coordinates of \(L\) and \(P\). Let's assume \(L=(x_1,y_1)\) and \(P=(x_2,y_2)\) from the grid. Suppose \(L=(4,3)\) and \(P=(7,1)\) and \(k = \frac{1}{3}\).

Step2: Calculate the \(x\) - coordinate of \(L'\)

Using the formula \(x_{l'}=x_p + k(x_l - x_p)\), we substitute \(x_p = 7\), \(k=\frac{1}{3}\), and \(x_l = 4\). So \(x_{l'}=7+\frac{1}{3}(4 - 7)=7+\frac{1}{3}\times(- 3)=7 - 1=6\).

Step3: Calculate the \(y\) - coordinate of \(L'\)

Using the formula \(y_{l'}=y_p + k(y_l - y_p)\), we substitute \(y_p = 1\), \(k=\frac{1}{3}\), and \(y_l = 3\). So \(y_{l'}=1+\frac{1}{3}(3 - 1)=1+\frac{2}{3}=\frac{3 + 2}{3}=\frac{5}{3}\approx1.67\).

Answer:

The coordinates of \(L'\) are \((6,\frac{5}{3})\) (coordinates are based on the assumed original coordinates of \(L\) and \(P\) from the grid. If actual coordinates are different, recalculate using the above method).