Question

the diagram shows a dilation of line ab about the origin o. determine the scale factor of the dilation by following these steps.

1. measure these lengths:

oa = units
oa = units

Explanation:

Step1: Measure the length of OA

Using the distance formula (or counting units on the grid). Point \(A\) is at \((2,-1)\) and \(O(0,0)\). The distance \(OA=\sqrt{(2 - 0)^2+(-1-0)^2}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\) (or by counting the units in the right - triangle formed with legs \(2\) and \(1\), we can also use the grid: if we consider the horizontal and vertical distances. The horizontal distance from \(O\) to \(A\) is \(2\) units and the vertical distance is \(1\) unit. Using the Pythagorean theorem \(OA=\sqrt{2^{2}+1^{2}}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\). But if we just count the "diagonal" units (assuming each square has side - length \(1\)) in a non - strict sense (for the purpose of dilation scale factor, since the ratio will be the same), we can note that from the grid, \(OA\) is composed of \(2\) horizontal and \(1\) vertical unit - lengths.

Step2: Measure the length of \(OA'\)

Point \(A'\) is at \((3,-1.5)\). The distance \(OA'=\sqrt{(3 - 0)^2+(-1.5-0)^2}=\sqrt{9+2.25}=\sqrt{11.25}=\frac{3\sqrt{5}}{2}\approx3.35\) (or using the non - strict grid counting: the horizontal distance from \(O\) to \(A'\) is \(3\) units and the vertical distance is \(1.5\) units. Using the Pythagorean theorem \(OA'=\sqrt{3^{2}+(1.5)^{2}}=\sqrt{9 + 2.25}=\sqrt{11.25}\). Another way: if we consider the ratio of corresponding coordinates. Since dilation about the origin \(O\) has the rule \((x,y)\to(kx,ky)\). For point \(A(2,-1)\) and \(A'(3,-1.5)\), we can find \(k\) by looking at the \(x\) - coordinates (or \(y\) - coordinates). \(\frac{3}{2}=1.5\) (from \(x\) - coordinates: \(x = 2\) for \(A\) and \(x'=3\) for \(A'\)) or \(\frac{-1.5}{-1}=1.5\) (from \(y\) - coordinates).

Answer:

\(OA = \sqrt{5}\approx2.24\) units, \(OA'=\frac{3\sqrt{5}}{2}\approx3.35\) units. The scale factor \(k = 1.5\)