Two-dimensional Shapes
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Texas 8th Grade Math › Two-dimensional Shapes
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 $k$.
Consider these examples: a hexagon is rotated $90^\circ$ clockwise; a triangle is slid 4 units right; a square is reflected across the $x$-axis; and a star is dilated by a factor of 2. Which transformations preserve orientation?
Reflections and translations
Rotations and translations
Reflections and dilations
Reflections only
Explanation
Orientation means the order/direction of the vertices (clockwise vs. counterclockwise) stays the same. Rotations and translations keep the figure facing the same way, so they preserve orientation (and congruence). Reflections flip the figure, reversing orientation (though they keep congruence). Dilations keep orientation but change size, so they do not preserve congruence unless the scale factor is 1.
Reflect $P(3,-2)$ over the $x$-axis. What are the new coordinates?
$(-3,2)$
$(3,2)$
$(3,-2)$
$(-3,-2)$
Explanation
Reflection over the $x$-axis uses the rule $(x,y)\to(x,-y)$. Substitute $P(3,-2)$: $(3,-(-2))=(3,2)$. Geometrically, reflecting across the $x$-axis keeps $x$ the same and flips the sign of $y$, moving the point straight up or down.
Rotate $A(2,5)$ $90^\circ$ clockwise about the origin. What are the new coordinates?
$(-5,2)$
$(-2,-5)$
$(-5,-2)$
$(5,-2)$
Explanation
A $90^\circ$ clockwise rotation uses $(x,y)\to(y,-x)$. Substitute $(2,5)$: $(5,-2)$. Geometrically, the coordinates swap and the original $x$ becomes negative.
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.
Which transformations preserve congruence (same size and shape)? For example, think about a triangle rotated $180^\circ$, a square slid left, a pentagon reflected across the $y$-axis, and a hexagon dilated by factor 2.
Dilations only
Reflections only
Rotations and dilations
Rotations, translations, and reflections
Explanation
Congruence means same size and shape. The rigid motions—rotations, translations, and reflections—preserve congruence. Dilations change size (unless the scale factor is 1), so they do not preserve congruence.
Points $Q(-4,1)\to Q'(4,1)$ and $R(3,-5)\to R'(-3,-5)$. Which rule describes this transformation?
$(x,y)\to(-x,y)$
$(x,y)\to(x,-y)$
$(x,y)\to(y,x)$
$(x,y)\to(-x,-y)$
Explanation
Both images change the sign of $x$ while $y$ stays the same, so the rule is $(x,y)\to(-x,y)$. For example, $(-4,1)\to(4,1)$ and $(3,-5)\to(-3,-5)$. This is a reflection across the $y$-axis.
Which option shows a transformation that preserves both orientation and congruence?
A triangle is rotated $180^\circ$ about the origin.
A parallelogram is reflected across the $y$-axis.
A square is dilated by scale factor $\tfrac{1}{2}$.
A pentagon is dilated by factor 3.
Explanation
Rotations and translations preserve both orientation (the figure's facing direction) and congruence (same size and shape). Reflections preserve congruence but reverse orientation. Dilations preserve orientation but change size, so they do not preserve congruence unless the scale factor is 1.
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 $=60\times4=240\ \text{cm}^2$.