Proportionality>Explaining Effects of Scale Factors on Figures Using Algebraic Representations(TEKS.Math.8.3.C)
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Texas 8th Grade Math › Proportionality>Explaining Effects of Scale Factors on Figures Using Algebraic Representations(TEKS.Math.8.3.C)
A triangle has vertices (2, 3), (6, 1), and (4, 7). It is dilated about the origin by a scale factor of 2.5. What are the coordinates of the image?
(5, 7.5), (15, 2.5), (10, 17.5)
(4, 6), (12, 2), (8, 14)
(4.5, 5.5), (8.5, 3.5), (6.5, 9.5)
(5, 3), (15, 1), (10, 7)
Explanation
For a dilation centered at the origin with scale factor k, use the rule $(x,y) \to (kx, ky)$. Here $k=2.5$. Compute each vertex: (2,3) → (2.5·2, 2.5·3) = (5, 7.5); (6,1) → (15, 2.5); (4,7) → (10, 17.5). Because each coordinate is multiplied by the same positive factor, side lengths scale by 2.5, angles stay the same, and the figure is an enlargement centered at the origin.
A rectangle with vertices (1, 2), (5, 2), (5, 6), and (1, 6) undergoes a dilation centered at the origin with scale factor 0.4. Which rule represents this dilation?
$(x,y) \to (x+0.4,, y+0.4)$
$(x,y) \to (0.4x,, 0.4y)$
$(x,y) \to (2.5x,, 2.5y)$
$(x,y) \to (0.4y,, 0.4x)$
Explanation
For a dilation about the origin with scale factor $k$, the rule is $(x,y) \to (kx, ky)$. With $k=0.4$, the correct rule is $(x,y) \to (0.4x, 0.4y)$. This multiplies both coordinates by the same positive rational factor, producing a reduction that preserves shape and proportional relationships.
A parallelogram has vertices at (-3, 2), (1, 2), (4, 6), and (0, 6). It is dilated by a scale factor of 1.5 about the origin. What are the coordinates of the image?
(-1.5, 1), (0.5, 1), (2, 3), (0, 3)
(-1.5, 3.5), (2.5, 3.5), (5.5, 7.5), (1.5, 7.5)
(-4.5, 3), (1.5, 3), (6, 9), (0, 9)
(-3, 3), (1, 3), (4, 9), (0, 9)
Explanation
Use $(x,y) \to (kx, ky)$ with $k=1.5$. Compute: (-3,2) → (-4.5,3), (1,2) → (1.5,3), (4,6) → (6,9), (0,6) → (0,9). The dilation multiplies all lengths by 1.5 and preserves parallelism and angle measures, keeping the shape while enlarging it from the origin.
Triangle ABC has vertices at (-2, -5), (3, -1), and (0, 4). The triangle is dilated about the origin by a scale factor of 0.75. What are the coordinates of the image?
(-1.25, -4.25), (3.75, -0.25), (0.75, 4.75)
(-0.5, -1.25), (0.75, -0.25), (0, 1)
(-1.5, -5), (2.25, -1), (0, 4)
(-1.5, -3.75), (2.25, -0.75), (0, 3)
Explanation
A dilation centered at the origin uses $(x,y) \to (kx, ky)$. With $k=0.75$: (-2,-5) → (-1.5,-3.75); (3,-1) → (2.25,-0.75); (0,4) → (0,3). Since $k<1$, the figure is a reduction; side lengths shrink by the same factor while angles and overall shape are preserved about the origin.
A right triangle has vertices at (2, -3), (-1, 4), and (0, -2). It is dilated by a scale factor of 3 about the origin. Which set lists the coordinates of the image?
(6, -9), (-3, 12), (0, -6)
(4, -6), (-2, 8), (0, -4)
(5, 0), (2, 7), (3, 1)
(6, -9), (12, -3), (0, -6)
Explanation
Apply the rule $(x,y) \to (kx, ky)$ with $k=3$. Compute: (2,-3) → (6,-9); (-1,4) → (-3,12); (0,-2) → (0,-6). Multiplying both coordinates by the same positive factor preserves proportional relationships and angle measures, producing a larger, similar triangle centered at the origin.