Proportionality>Comparing and Contrasting Dilations on a Coordinate Plane(TEKS.Math.8.3.B)
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Texas 8th Grade Math › Proportionality>Comparing and Contrasting Dilations on a Coordinate Plane(TEKS.Math.8.3.B)
Square ABCD has vertices A(0,0), B(4,0), C(4,4), D(0,4). After dilation with scale factor 1.5 centered at the origin, square A'B'C'D' has vertices A'(0,0), B'(6,0), C'(6,6), D'(0,6).
How do the perimeter and area of the dilated square compare to the original?
Perimeter is multiplied by 2.25; area is multiplied by 1.5
Perimeter stays the same; area is multiplied by 2.25
Perimeter is multiplied by 1.5; area is multiplied by 2.25
Perimeter is multiplied by 1.5; area stays the same
Explanation
A dilation with scale factor k affects linear measurements by ×k and area by ×k^2. Here k = 1.5, so perimeter scales by 1.5 and area scales by 1.5^2 = 2.25. Angle measures, shape, and parallel sides are preserved.
Rectangle WXYZ has vertices W(-6,2), X(2,2), Y(2,-2), Z(-6,-2). After dilation with scale factor 0.5 centered at the origin, W'(-3,1), X'(1,1), Y'(1,-1), Z'(-3,-1).
Which statement about attributes is true after the dilation?
Corresponding angle measures are equal and corresponding sides are parallel.
All side lengths stay the same, so perimeter is unchanged.
The area is multiplied by 0.5.
The slopes of the sides are multiplied by 0.5.
Explanation
Dilations preserve shape, angle measures, and parallel relationships. Linear dimensions (and perimeter) scale by k = 0.5, and area scales by k^2 = 0.25 (not 0.5). Because corresponding sides remain parallel, their slopes are unchanged.
Triangle PQR has vertices P(1,1), Q(5,1), R(1,4). After dilation with scale factor 3 centered at the origin, triangle P'Q'R' has vertices P'(3,3), Q'(15,3), R'(3,12).
What is the ratio of the area of triangle P'Q'R' to the area of triangle PQR?
3
6
27
9
Explanation
Area scales by the square of the scale factor. With k = 3, the area scale factor is k^2 = 9. Angle measures and shape are preserved, while side lengths and perimeter are multiplied by k.
Parallelogram JKLM has vertices J(2,1), K(6,1), L(7,4), M(3,4). After dilation with scale factor 0.75 centered at the origin, J', K', L', M' are the images.
How do the slopes of corresponding sides compare after the dilation?
Each slope is multiplied by 0.75.
Slopes remain the same for corresponding sides.
Each slope is divided by 0.75.
Each slope becomes its negative reciprocal.
Explanation
Dilations are similarity transformations that preserve angle measures and parallel lines. Corresponding sides stay parallel, so their slopes are unchanged. Linear measures scale by k = 0.75 and area by k^2, but slopes do not scale.
Pentagon ABCDE has vertices A(1,1), B(4,1), C(5,3), D(3,5), E(1,3). After dilation with scale factor 2 centered at the origin, pentagon A'B'C'D'E' is the image.
Which statement correctly compares attributes of the original and dilated pentagons?
The perimeter doubles, and all angle measures stay the same.
The perimeter doubles, and all angle measures double.
The perimeter stays the same, and all angle measures stay the same.
The perimeter quadruples, and all angle measures stay the same.
Explanation
With scale factor k = 2, linear measurements (including perimeter) are multiplied by 2, while area is multiplied by k^2 = 4. Dilations preserve shape and angle measures, and keep corresponding sides parallel.