Question

1. here is a polygon on a grid.

a. draw a scaled copy of the polygon using a scale factor 3. label the copy a.
b. draw a scaled copy of the polygon with a scale factor \\( \frac { 1 } { 2 } \\). label it b.
c. is polygon a a scaled copy of polygon b? if so, what is the scale factor that takes b to a?

Explanation:

Step1: Determine the coordinates of the original polygon's vertices

Assume the original polygon has vertices at certain grid - points. For example, if we consider a simple polygon (the exact coordinates depend on the figure in the grid, but for illustration, let's say we have a polygon with vertices \( (x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n) \))

Step2: Scale for polygon A

For a scale factor of \( k = 3 \), the new coordinates of the vertices of polygon A will be \( (3x_1,3y_1),(3x_2,3y_2),\cdots,(3x_n,3y_n) \). Plot these points on the grid and connect them to form polygon A.

Step3: Scale for polygon B

For a scale factor of \( k=\frac{1}{2} \), the new coordinates of the vertices of polygon B will be \( (\frac{1}{2}x_1,\frac{1}{2}y_1),(\frac{1}{2}x_2,\frac{1}{2}y_2),\cdots,(\frac{1}{2}x_n,\frac{1}{2}y_n) \). Plot these points on the grid and connect them to form polygon B.

Step4: Check if A is a scaled copy of B

Let the scale factor from B to A be \( k\). If the coordinates of B are \( (b_1,b_2,\cdots,b_n) \) and of A are \( (a_1,a_2,\cdots,a_n) \), then \( k=\frac{a_i}{b_i}\) (for \(i = 1,\cdots,n\)). Since \(a_i = 3x_i\) and \(b_i=\frac{1}{2}x_i\), then \(k=\frac{3x_i}{\frac{1}{2}x_i}=6\)

Answer:

a. and b. (Drawings are done by scaling coordinates as described above). c. Yes, Polygon A is a scaled copy of Polygon B. The scale factor is \(6\)