Question
Suppose line k has equation xequals3. Compare the areas of ABCD and Upper A prime Upper B prime Upper C prime Upper D primeequalsleft parenthesis Upper T Subscript left angle 1 comma 2 right angle Baseline circle Upper R Subscript k right parenthesis left parenthesis ABCD right parenthesis.
Question content area bottom
Part 1
Suppose ABCD is a rectangle with length l and width w. Its area is given by the expression
enter your response here. Rigid motions
▼
sometimes
never
always
preserve lengths and widths, so they
▼
always
sometimes
never
change the value of the area. Therefore, the areas of ABCD and Upper A prime Upper B prime Upper C prime Upper D prime
▼
must
cannot
might or might not
be equal. Similar reasoning applies when ABCD is not a rectangle.
Question content area bottom
Part 1
Suppose ABCD is a rectangle with length l and width w. Its area is given by the expression
enter your response here. Rigid motions
▼
sometimes
never
always
preserve lengths and widths, so they
▼
always
sometimes
never
change the value of the area. Therefore, the areas of ABCD and Upper A prime Upper B prime Upper C prime Upper D prime
▼
must
cannot
might or might not
be equal. Similar reasoning applies when ABCD is not a rectangle.
Answers
GPT-4o mini
Let's analyze the problem step by step:
1. **Area of Rectangle ABCD**: The area \(A\) of rectangle ABCD is given by the formula:
\[
A = l \times w
\]
where \(l\) is the length and \(w\) is the width of the rectangle.
2. **Rigid Motions**: Rigid motions (such as translations, rotations, and reflections) are transformations that preserve distances and angles. Therefore, they do not change the lengths and widths of the shape upon transformation.
3. **Effect on Area**: Since rigid motions preserve lengths and widths, the area of a figure such as ABCD remains unchanged after the transformation. This means that the area of the transformed rectangle \(A'B'C'D'\) will also be given by the same formula:
\[
A' = l \times w
\]
Hence, we conclude that the area is not affected by rigid motions.
Based on this analysis, in response to each of the fill-in-the-blank prompts:
1. The area of rectangle ABCD is given by the expression \(l \times w\).
2. Rigid motions **always** preserve lengths and widths, so they **never** change the value of the area.
3. Therefore, the areas of ABCD and \(A'B'C'D'\) **must** be equal.
This reasoning remains valid even if ABCD is not a rectangle, as long as the transformation is a rigid motion, since such transformations will still preserve the distances and, hence, the area will remain the same.
1. **Area of Rectangle ABCD**: The area \(A\) of rectangle ABCD is given by the formula:
\[
A = l \times w
\]
where \(l\) is the length and \(w\) is the width of the rectangle.
2. **Rigid Motions**: Rigid motions (such as translations, rotations, and reflections) are transformations that preserve distances and angles. Therefore, they do not change the lengths and widths of the shape upon transformation.
3. **Effect on Area**: Since rigid motions preserve lengths and widths, the area of a figure such as ABCD remains unchanged after the transformation. This means that the area of the transformed rectangle \(A'B'C'D'\) will also be given by the same formula:
\[
A' = l \times w
\]
Hence, we conclude that the area is not affected by rigid motions.
Based on this analysis, in response to each of the fill-in-the-blank prompts:
1. The area of rectangle ABCD is given by the expression \(l \times w\).
2. Rigid motions **always** preserve lengths and widths, so they **never** change the value of the area.
3. Therefore, the areas of ABCD and \(A'B'C'D'\) **must** be equal.
This reasoning remains valid even if ABCD is not a rectangle, as long as the transformation is a rigid motion, since such transformations will still preserve the distances and, hence, the area will remain the same.