In this section, we explore a type of transformation which changes the distance between points. Before introducing some examples on the Cartesian \(xy\)-plane, we review how the Pythagorean Theorem is used to compute distance between points.
To compute the length of a segment on a graph we use the Pythagorean Theorem, noting that it is easy to compute distances along horizontal or vertical lines on a graph. To measure the distance between two points that not on a horizontal or vertical line, we construct a right triangle whose hypotenuse is the segment connecting the points. One leg of this triangle will be horizontal, the other vertical.
In Figure 5.1.1, the coordinates of point \(A\) are \((6,6)\) and of point \(B\) are \((9,2)\text{.}\) We think of the segment \(\overline{AB}\) as the hypotenuse of a triangle with a right angle at \(C=(6,2)\text{.}\) The length of the horizontal leg is found by subtracting the \(x\)-coordinates, \(m(\overline{BC})=9-6 = 3\text{.}\) Similarly, the vertical distance by subtracting the \(y\)-coordinates, \(m(\overline{AC})=6-2 = 4\text{.}\) Since segment \(\overline{AB}\) is the hypotenuse of the right triangle \(\Delta ABC\text{,}\) we can use the Pythagorean Theorem to find its length, namely \(m(\overline{AB})=\sqrt{4^2+3^2}=\sqrt{9+16}=\sqrt{25}=5\text{.}\)
To use the ‘Angle’ tool in GeoGebra, select a point on a side of the angle, followed by the vertex, and then by a point on the other side. Note that GeoGebra assumes that angles open in a counterclockwise direction. If you get the measure of an exterior angle, remeasure the angle selecting the points in the opposite direction.
Measure the new angles \(\angle A'B'C'\text{,}\)\(\angle B'C'D'\text{,}\)\(\angle C'D'A'\text{,}\) and \(\angle D'A'B'\text{.}\) How do they compare to the original four angles?
Let \(O\) be the point \((0,0)\text{.}\) Draw the rays \(\overrightarrow{OA'}\text{,}\)\(\overrightarrow{OB'}\text{,}\)\(\overrightarrow{OC'}\text{,}\) and \(\overrightarrow{OD'}\text{.}\) What do you notice about these rays?
In Task 5.1.1.a.iii, you constructed the image of quadrilateral \(ABCD\) under a geometric transformation that doubled the sides of the figure. We call this type of transformation a dilation. Checkpoint 5.1.3 encourages you to reflect on the key ideas that appeared in Exploration 5.1.1. These properties will be useful as we work through this chapter.
What does the scale factor tell you about the relationship between the lengths of segments in the dilated image compared to the lengths of segments in the original figure?
What is the image of quadrilateral \(A'B'C'D'\text{,}\) created in Exploration 5.1.1 under the dilation with center \((0,0)\) and scale factor \(\frac{1}{2}\text{?}\)
A dilation with center \(Q\) and scale factor \(k\) is a geometrical transformation that maps each point \(P\) to a unique point \(P'\) satisfying the following:
\(P'\) lies on line \(\overleftrightarrow{QP}\) and
The dilation in Figure 5.1.6 has a scale factor of \(1.75\) and center \(Q\text{.}\) Notice how \(A'\) lies on \(\overleftrightarrow{QA}\) and \(1.75(QA)=1.75(9)=15.75=QA'\text{.}\)
You can avoid using the Pythagorean Theorem by thinking of sides and diagonals of blocks on the grid as units; for example \(HF\) is equal to the diagonal of a \(2\times 4\)-rectangle so \(HF'\) should be equal to three of these diagonals.
We have observed that a dilation maps each figure onto a figure that has the same shape as the original, but the size is not preserved. When two figures have the same shape we say that the figures are similar. Corresponding angles will be congruent and lengths of corresponding sides will be proportional. We give a formal definition for similar triangles but note that this definition can easily be generalized for polygons with more than three sides.
We use the symbol \(\sim\) to denote similar figures. Thus, \(\Delta ABC\sim\Delta DEF\) means that \(\Delta ABC\) and \(\Delta DEF\) are similar triangles.
In Exploration 5.1.3, we consider a situation in which a vertex of a figure is mapped to itself by a dilation. As you complete this activity, look for relationships between similarity and dilation.
In Figure 5.1.10, \(\Delta A'B'C'\) is the image of \(\Delta ABC\) under a dilation with center \(D\text{.}\) Given that \(AD=4.42\text{,}\)\(A'D=1.77\text{,}\)\(BD=5.11\text{,}\) and \(C'D=2.05\text{,}\) determine the following:
