Exploration 5.1.1. Constructing a Dilated Quadrilateral.
The following tasks refer to Figure 5.1.2. This exploration may be completed virtually or on paper.
(a) Measuring a Quadrilateral.
(i)
Compute the lengths of each segment using the Pythagorean Theorem:
(A)
\(\overline{AB}\)
(B)
\(\overline{BC}\)
(C)
\(\overline{CD}\)
(D)
\(\overline{AD}\)
(ii)
Use either a protractor or GeoGebra tools to measure the angles:
(A)
\(\angle ABC\)
Hint.
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.
(B)
\(\angle BCD\)
(C)
\(\angle CDA\)
(D)
\(\angle DAB\)
(iii) Dilating the Quadrilateral.
Multiply the coordinates of each vertex by 2 to obtain coordinates for points \(A'\text{,}\) \(B'\text{,}\) \(C'\text{,}\) and \(D'\text{.}\)
Then plot and connect the new points to form quadrilateral \(A'B'C'D'\text{.}\)
(b) Relationships between the Image and the Original.
Refer to your construction in Figure 5.1.2 as you answer the following:
(i)
Describe how the quadrilateral \(A'B'C'D'\) you drew in Figure 5.1.2 relates to the original quadrilateral \(ABCD\text{.}\)
Hint.
This task is intentionally vague and open-ended.
(ii)
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?
(iii)
Compute the lengths of the following segments
(A)
\(\overline{A'B'}\)
(B)
\(\overline{B'C'}\)
(C)
\(\overline{C'D'}\)
(D)
\(\overline{A'D'}\)
Compare these to the original side lengths.
(iv)
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?
