Question

observing properties of dilations
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
check

Explanation:

Step1: Calculate the length of OA

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(O(0,0)\) and \(A(2,-1)\), \(OA=\sqrt{(2 - 0)^2+(-1 - 0)^2}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\) units. But if we use the grid - counting (assuming each grid square has side - length 1), from \(O(0,0)\) to \(A(2,-1)\), we can use the Pythagorean theorem: \(OA=\sqrt{2^{2}+1^{2}}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\) (or we can count the number of units in terms of the right - triangle sides formed by the coordinates). Another way is to note that if we consider the horizontal and vertical displacements. The horizontal displacement from \(O\) to \(A\) is \(2\) units and the vertical displacement is \(1\) unit. Using the Pythagorean theorem \(OA=\sqrt{2^{2}+1^{2}}=\sqrt{5}\approx2.24\). But if we assume the problem is using a non - coordinate - based length measurement (counting the number of units in a right - triangle - side - equivalent way on the grid), we can also use the fact that for a dilation, we can use the ratio of corresponding segments. However, if we use the coordinate formula: \(OA=\sqrt{(2 - 0)^2+(-1 - 0)^2}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\). But if we use the "counting" method (assuming each small square has side length \(1\)):

$$OA=\sqrt{2^{2}+1^{2}}=\sqrt{5}\approx2.24$$

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

For \(O(0,0)\) and \(A'(3,-1.5)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(OA'=\sqrt{(3 - 0)^2+(-1.5 - 0)^2}=\sqrt{9 + 2.25}=\sqrt{11.25}=\frac{3\sqrt{5}}{2}\approx3.35\). Using the Pythagorean theorem (horizontal displacement \(3\), vertical displacement \(1.5\)): \(OA'=\sqrt{3^{2}+1.5^{2}}=\sqrt{9 + 2.25}=\sqrt{11.25}=\frac{3\sqrt{5}}{2}\approx3.35\)

Answer:

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