Question

math 7 unit 1 cfu a version 2
name:___________________________
date:________ period:__ table:____
directions: justify your reasoning by showing all your work and thinking.
write your answers on the lines provided.

problem 1
which of parallelograms a, b, c, and d are scaled copies of parallelogram o?

image of grid with parallelograms o, a, b, c, d

a. scaled copies (select all that apply)
☐ a
☐ b
☐ c
☐ d
☐ none are scaled copies

b. choose one shape that is not a scaled copy and explain how you know.
use academic language to receive full credit.
_____________________________________________________
_____________________________________________________
_____________________________________________________

problem 2
for each figure and its copy, explain what you know about the scale factor. use academic language to
receive full credit.

image of grid with rectangles (original and copies) for part a
a)

image of grid with rectangles (original and copies) for part b
b)

Answer:

Problem 1a

To determine scaled copies, we check if the side - length ratios and angle measures (for parallelograms, angles should be preserved) are consistent with a scale factor. Let's assume the grid has unit squares.

1. Analyze parallelogram O: Let's find its base and height (or side lengths and the slant height). Suppose the base of O is \(b_O\) and the height (or the length of the non - base side with respect to the grid) is \(h_O\).
2. Analyze parallelogram A: Calculate the ratio of its base to O's base and its non - base side to O's non - base side. If the ratios are equal, it's a scaled copy.
3. Analyze parallelogram B: Do the same as for A. Check the ratio of corresponding sides.
4. Analyze parallelogram C: Check the angle. A scaled copy should have the same angle as the original. If the angle of C is different from O (e.g., if the slope of the non - base side is different), it's not a scaled copy.
5. Analyze parallelogram D: Check the side - length ratios.

After analyzing (assuming standard grid - based analysis):

• Parallelogram A: The ratio of base and the non - base side (slant side) is the same as O. So it's a scaled copy.
• Parallelogram B: The ratio of base and the non - base side (slant side) is the same as O. So it's a scaled copy.
• Parallelogram D: The ratio of base and the non - base side (slant side) is the same as O. So it's a scaled copy.
• Parallelogram C: The angle of the non - base side (slant side) with respect to the base is different from O. So it's not a scaled copy.

So the scaled copies are A, B, D.

Problem 1b

Let's choose parallelogram C. A scaled copy of a parallelogram must have the same angle measures (since scaling preserves the shape, i.e., the angles) and the ratios of corresponding side lengths must be equal (the scale factor). For parallelogram O and C, the slope of the non - base (slant) side is different. This means the angle between the base and the slant side is different. Also, when we calculate the ratio of the length of the slant side to the base side for O and C, the ratios are not equal. So, since the angle is not preserved and the side - length ratios are not equal, C is not a scaled copy of O.

Problem 2a

1. Let's assume the original rectangle has length \(l_o\) and width \(w_o\), and the copy (the smaller rectangle above) has length \(l_1\) and width \(w_1\), and the other copy (the smaller rectangle below) has length \(l_2\) and width \(w_2\).
2. First, find the scale factor for the upper copy: Calculate the ratio of the length of the copy to the original length (\(s_1=\frac{l_1}{l_o}\)) and the ratio of the width of the copy to the original width (\(s_2 = \frac{w_1}{w_o}\)). If \(s_1=s_2\), then it's a scaled copy. Let's say the original length is, for example, 6 units and the original width is 4 units. If the upper copy has length 4 units and width \(\frac{8}{3}\) units (assuming grid - based measurement), \(\frac{4}{6}=\frac{2}{3}\) and \(\frac{\frac{8}{3}}{4}=\frac{2}{3}\), so the scale factor is \(\frac{2}{3}\) (since the copy is smaller than the original, the scale factor is less than 1).
3. For the lower copy: Calculate the ratio of its length to the original length and its width to the original width. If the original width is 4 units and the lower copy's width is 2 units, and the original length is 6 units and the lower copy's length is 4 units. \(\frac{4}{6}=\frac{2}{3}\) and \(\frac{2}{4}=\frac{1}{2}\). Since \(\frac{2}{3}

eq\frac{1}{2}\), the lower copy is not a scaled copy (it is a distorted version, not a scaled one). The upper copy has a scale factor of \(\frac{2}{3}\) (scale factor less than 1, so it's a reduction).

Problem 2b

1. Let the original rectangle have length \(l_o\) and width \(w_o\), and the larger copy have length \(l_1\) and width \(w_1\), and the smaller copy (the one below the larger) have length \(l_2\) and width \(w_2\).
2. For the larger copy: Calculate the ratio of its length to the original length (\(s_1=\frac{l_1}{l_o}\)) and the ratio of its width to the original width (\(s_2=\frac{w_1}{w_o}\)). If \(s_1 = s_2>1\), it's an enlargement. Let's assume the original length is 4 units and the original width is 3 units. If the larger copy has length 8 units and width 6 units, \(\frac{8}{4}=2\) and \(\frac{6}{3}=2\), so the scale factor is 2 (enlargement, scale factor greater than 1).
3. For the smaller copy (below the larger one): Calculate the ratio of its length to the original length and its width to the original width. If the original length is 4 units and the smaller copy's length is 4 units, and the original width is 3 units and the smaller copy's width is 1 unit. \(\frac{4}{4}=1\) and \(\frac{1}{3}

eq1\). So it's not a scaled copy (distorted), while the larger one has a scale factor of 2.

Final Answers

Problem 1a

A, B, D

Problem 1b

(Example) Parallelogram C is not a scaled copy. A scaled copy must have equal ratios of corresponding side lengths and preserved angles. For O and C, the angle between the base and the slant side differs, and side - length ratios are unequal, so C is not a scaled copy.

Problem 2a

(Example) The upper smaller rectangle is a scaled copy of the original with a scale factor of \(\frac{2}{3}\) (reduction), while the lower smaller rectangle is not a scaled copy (side - length ratios are unequal).

Problem 2b

(Example) The larger rectangle is a scaled copy of the original with a scale factor of 2 (enlargement), while the smaller rectangle below the larger one is not a scaled copy (side - length ratios are unequal).