Question

8 if a rectangle has dimensions of 10 meters by 5 meters and is scaled by a factor of 2, what are the new dimensions?
a 5 meters by 2.5 meters
b 15 meters by 10 meters
c 20 meters by 10 meters
d 30 meters by 15 meters

1. which statement about similar circles is always true?

a their diameters are equal.
b their radii are equal.
c they have the same area.
d their corresponding radii are proportional.

1. a circle is dilated with a scale factor of 3 and moved so its center aligns with another circle center. what transformations were applied?

a translation and reflection
b dilation and reflection
c dilation and translation
d rotation and dilation

1. a smaller circle with a radius of 3 and center (-2, 4) needs to be transformed into a larger circle with a radius of 10 and center (3, -2). what transformations are needed?

a dilation only
b translation only
c reflection and dilation
d dilation and translation

1. a bridge is 300 meters wide and is being scaled down by a factor of 0.02 for a model. what is the width of the model of the bridge?

a 6 meters
b 4 meters
c 7 meters
d 5 meters

1. a square with vertices at (1, 1), (1, 3), (3, 3), and (3, 1) is dilated with a scale factor of 2 using the origin as the center. what are the coordinates of the new vertices?

a (1, 2), (1, 4), (3, 4), (3, 2)
b (0.5, 0.5), (0.5, 1.5), (1.5, 1.5), (1.5, 0.5)
c (2, 2), (2, 6), (6, 6), (6, 2)
d (2, 2), (2, 6), (6, 6), (6, 2)

1. a rectangular park is 600 meters long and 400 meters wide. using a scale factor of 0.05, what will be the width of the scaled - down park?

a 10 meters
b 20 meters
c 25 meters
d 40 meters

Explanation:

Step1: Scale the rectangle's dimensions

When a rectangle is scaled by a factor \(k\), the new length \(L_{new}=k\times L_{old}\) and new width \(W_{new}=k\times W_{old}\). Given \(L_{old} = 10\) meters, \(W_{old}=5\) meters, and \(k = 2\).
For the length: \(L_{new}=2\times10=20\) meters.
For the width: \(W_{new}=2\times5 = 10\) meters.

Step2: Analyze similar - circle properties

Similar circles (in fact, all circles are similar) have the property that their corresponding radii (and also diameters) are proportional. If two circles have radii \(r_1\) and \(r_2\), the ratio \(\frac{r_1}{r_2}\) is a constant (the scale factor). Equal radii (or diameters) or equal areas would mean the circles are congruent (a special case of similar with scale factor \(k = 1\)), but similarity only requires proportional radii.

Step3: Identify circle - transformation types

Dilation changes the size of a figure (in this case, a circle, changing its radius). Translation moves the figure (in this case, moving the center of the circle to another position). Reflection flips a figure over a line (not relevant here as we are only changing size and position). So, dilating a circle (changing its radius with a scale factor) and moving its center (translation) are the transformations.

Step4: Determine transformations for circle - size and position change

The radius changes from \(r_1 = 2\) to \(r_2=10\) (a dilation with scale factor \(k=\frac{10}{2}=5\)). The center changes from \((-2,4)\) to \((3,-2)\). To move the center from \((x_1,y_1)\) to \((x_2,y_2)\), we use the translation rule \((x,y)\to(x+(x_2 - x_1),y+(y_2 - y_1))\). There is no reflection (flipping) involved, so the transformations are dilation (to change the size) and translation (to change the center's position).

Step5: Calculate the model - bridge width

When scaling down, if the original width \(W_{old}=300\) meters and the scale factor \(k = 0.02\), then \(W_{new}=k\times W_{old}\).
\(W_{new}=0.02\times300=6\) meters.

Step6: Find dilated - square vertices

When dilating a point \((x,y)\) with a scale factor \(k\) about the origin \((0,0)\), the new point is \((kx,ky)\). For the square with vertices \((1,1)\), \((1,2)\), \((2,2)\), \((2,1)\) and \(k = 2\):

• For \((1,1)\): \((2\times1,2\times1)=(2,2)\)
• For \((1,2)\): \((2\times1,2\times2)=(2,4)\)
• For \((2,2)\): \((2\times2,2\times2)=(4,4)\)
• For \((2,1)\): \((2\times2,2\times1)=(4,2)\)

Step7: Calculate the scaled - down park width

Given the original width of the park \(W_{old}=400\) meters and scale factor \(k = 0.05\). Using the formula \(W_{new}=k\times W_{old}\).
\(W_{new}=0.05\times400 = 20\) meters.

Answer:

1. c. 20 meters by 10 meters
2. d. Their corresponding radii are proportional
3. c. Dilation and translation
4. d. Dilation and translation
5. a. 6 meters
6. c. \((2,2),(2,4),(4,4),(4,2)\)
7. c. 20 meters