Two-dimensional Shapes>Modeling the Effect of Dilations on Linear and Area Measurements(TEKS.Math.8.10.D)
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Texas 8th Grade Math › Two-dimensional Shapes>Modeling the Effect of Dilations on Linear and Area Measurements(TEKS.Math.8.10.D)
Rectangle ABCD has length 8 cm and width 6 cm. After a dilation with scale factor 1.5, rectangle A'B'C'D' is created. Which option correctly gives the new linear dimensions and area?
Length 12 cm, width 9 cm; area 72 $\text{cm}^2$
Length 12 cm, width 9 cm; area 108 $\text{cm}^2$
Length 12 cm, width 9 cm; area 144 $\text{cm}^2$
Length 10 cm, width 7.5 cm; area 108 $\text{cm}^2$
Explanation
Dilations multiply all linear measurements by $k$ and all area measurements by $k^2$. Here $k=1.5$, so length $8\to 8\times1.5=12$ and width $6\to 6\times1.5=9$. Original area $=8\times6=48$. New area $=48\times(1.5)^2=48\times2.25=108$. Understanding $k$ vs. $k^2$ prevents mistakes like scaling area only by $k$.
A triangle has perimeter 24 units and area 18 square units. It is dilated by scale factor 0.5. What are the new perimeter and area?
Perimeter 12 units; area 9 square units
Perimeter 24 units; area 18 square units
Perimeter 12 units; area 36 square units
Perimeter 12 units; area 4.5 square units
Explanation
All linear measures (like perimeter) are multiplied by $k$, and area is multiplied by $k^2$. With $k=0.5$, the new perimeter is $24\times0.5=12$ units, and the new area is $18\times(0.5)^2=18\times0.25=4.5$ square units.
A circle has radius 10 cm. After a dilation with scale factor 0.8, what happens to its circumference and area?
Circumference is multiplied by 0.8; area is multiplied by 0.64
Circumference is multiplied by 0.64; area is multiplied by 0.8
Both circumference and area are multiplied by 0.8
Circumference stays the same; area is multiplied by 0.64
Explanation
Linear measurements (radius, diameter, and circumference) scale by $k=0.8$, while area scales by $k^2=(0.8)^2=0.64$. For a numeric check: original $C=20\pi$, new $C=16\pi$ (factor 0.8); original $A=100\pi$, new $A=64\pi$ (factor 0.64).
A parallelogram has base 12 cm and height 5 cm. It is dilated by a scale factor of 2 to create a similar parallelogram. Which statement is true?
Base 24 cm, height 10 cm; area 120 $\text{cm}^2$
Base 24 cm, height 10 cm; area 60 $\text{cm}^2$
Base 24 cm, height 10 cm; area 240 $\text{cm}^2$
Base 12 cm, height 5 cm; area 240 $\text{cm}^2$
Explanation
Linear measures scale by $k=2$: base $12\to24$, height $5\to10$. Area scales by $k^2=4$. Original area $=12\times5=60\ \text{cm}^2$, so new area $=60\times4=240\ \text{cm}^2$.
A square has side length 7 m. It is dilated by a scale factor of $\frac{1}{3}$. What are the new side length and area?
Side $\tfrac{7}{9}$ m; area $\tfrac{49}{3}$ $\text{m}^2$
Side $\tfrac{7}{3}$ m; area $\tfrac{49}{9}$ $\text{m}^2$
Side $\tfrac{7}{3}$ m; area $\tfrac{49}{3}$ $\text{m}^2$
Side 7 m; area $\tfrac{49}{9}$ $\text{m}^2$
Explanation
Linear measurements scale by $k=\tfrac{1}{3}$, so the side becomes $7\times\tfrac{1}{3}=\tfrac{7}{3}$ m. Area scales by $k^2=\left(\tfrac{1}{3}\right)^2=\tfrac{1}{9}$, so the area becomes $49\times\tfrac{1}{9}=\tfrac{49}{9}\ \text{m}^2$.