Respan Dataset Explorer

The figure on the grid is the result of enlarging a rectangle by a scale factor of 1.5.

On a coordinate plane, a rectangle has a length of 12 and width of 3.
Which graph shows the original rectangle?
On a coordinate plane, a rectangle has a length of 18 and width of 4.
On a coordinate plane, a rectangle has a length of 8 and width of 2.
On a coordinate plane, a rectangle has a length of 8 and width of 3.
On a coordinate plane, a rectangle has a length of 12 and width of 3.
Mark this and return

To determine which graph shows the original rectangle before it was enlarged by a scale factor of 1.5, we need to find the dimensions of the original rectangle based on the information given after it was enlarged.

### Given:
- The enlarged rectangle has a length of 12 units and a width of 3 units.
- The scale factor for the enlargement is 1.5.

### Finding the original dimensions:
To find the dimensions of the original rectangle, we divide the dimensions of the enlarged rectangle by the scale factor:

1. **Original Length:**
   \[
   \text{Original Length} = \frac{\text{Enlarged Length}}{\text{Scale Factor}} = \frac{12}{1.5} = 8 \text{ units}
   \]

2. **Original Width:**
   \[
   \text{Original Width} = \frac{\text{Enlarged Width}}{\text{Scale Factor}} = \frac{3}{1.5} = 2 \text{ units}
   \]

### Conclusion:
The original dimensions of the rectangle are 8 units for the length and 2 units for the width.

Among the choices provided:
- **On a coordinate plane, a rectangle has a length of 18 and width of 4.** (Incorrect)
- **On a coordinate plane, a rectangle has a length of 8 and width of 2.** (Correct)
- **On a coordinate plane, a rectangle has a length of 8 and width of 3.** (Incorrect)
- **On a coordinate plane, a rectangle has a length of 12 and width of 3.** (Incorrect)

The correct answer is: **On a coordinate plane, a rectangle has a length of 8 and width of 2.**

The number of calories in 5 cups of yogurt is 750. What proportion can you set up to determine the number of calories, c,  in 9 cups of yogurt?
StartFraction 5 over 750 EndFraction = StartFraction 9 over c EndFraction
StartFraction 9 over 750 EndFraction = StartFraction 5 over c EndFraction
StartFraction 750 over 5 EndFraction = StartFraction 9 over c EndFraction
StartFraction 750 over 9 EndFraction = StartFraction 5 over c EndFraction
Mark this and return

To determine the number of calories \( c \) in 9 cups of yogurt using the given information about 5 cups containing 750 calories, we want to set up a proportion based on the number of calories per cup.

1. **Given**: 
   - 5 cups of yogurt = 750 calories.
   - We want to find the calories in 9 cups \( c \).

Using the ratio of cups to calories, we can set up the proportion as:

\[
\frac{5}{750} = \frac{9}{c}
\]

Now let's evaluate the provided options based on this setup:

1. **\(\frac{5}{750} = \frac{9}{c}\)** 
   - This is the correct proportion.

2. **\(\frac{9}{750} = \frac{5}{c}\)**
   - This is incorrect; it mixes the positions incorrectly.

3. **\(\frac{750}{5} = \frac{9}{c}\)**
   - This is not the correct relationship in this context.

4. **\(\frac{750}{9} = \frac{5}{c}\)**
   - This again does not reflect the correct setup for the proportions based on the information given.

### Conclusion:
The valid proportion for finding the number of calories \( c \) in 9 cups of yogurt is:

**\(\frac{5}{750} = \frac{9}{c}\)**.

If 20 Sunday newspapers weigh 18 pounds, how many pounds do 30 Sunday newspapers weigh?
27 pounds
28 pounds
33 pounds
38 pounds
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To determine how many pounds 30 Sunday newspapers weigh, we can first calculate the weight of one Sunday newspaper and then use that information to find the total weight for 30 newspapers.

1. **Calculate the weight of one Sunday newspaper**:
   - If 20 Sunday newspapers weigh 18 pounds, then the weight of one newspaper is:
   \[
   \text{Weight of one newspaper} = \frac{18 \, \text{pounds}}{20} = 0.9 \, \text{pounds}
   \]

2. **Calculate the weight of 30 Sunday newspapers**:
   - Now, we multiply the weight of one newspaper by the number of newspapers (30):
   \[
   \text{Weight of 30 newspapers} = 30 \times 0.9 \, \text{pounds} = 27 \, \text{pounds}
   \]

### Conclusion:
The weight of 30 Sunday newspapers is **27 pounds**.

The regular pentagon below is a scale drawing of an original pentagon that was made by using a scale factor of One-fourth.
A hexagon with side lengths of 5.5 inches.

If each side of the pentagon has the same measure, which applies to the original pentagon? Select three options.
The scale drawing represents a reduction.
Each side of the original is 22 inches long.
One side of the model can be 1.375 inches.
The scale drawing is proportional to the original.
The scale factor is multiplied by 5.5 to get the dimensions of the original.
Mark this and return

To analyze the situation regarding the regular pentagon and its original dimensions, we can start with the information provided:

1. The pentagon is a scale drawing using a scale factor of **One-fourth**, meaning the original pentagon is larger than the scale drawing.
2. If we assume the **hexagon with side lengths of 5.5 inches** is relevant for comparison to show some conversions, let’s clarify:

### To determine the original pentagon's side length:

If the scale factor is **One-fourth**, the original dimensions will be **4 times** larger than those in the scale drawing. 

Suppose we refer back to the pentagon and to find its sides if each side of the pentagon is considered \(x\):

\[ 
\text{Original Side Length} = \text{Scale Drawn Side Length} \times \text{Scale Factor} \\
= x \times 4
\]

### Evaluating the Options:

1. **The scale drawing represents a reduction.**
   - **Correct**: The scale drawing is smaller than the original pentagon due to the scale factor of one-fourth.

2. **Each side of the original is 22 inches long.**
   - **Incorrect unless the pentagon's scale shortened from a larger side; therefore, only if backtracked would solidify depending on sides might assume the same lengths. If 5.5 assumed it relates to the hexagon and reduces; but for a defined pentagon needing clarity verifies an output remain something like: 22 inches would be reflected from original \(x\), thus set to multiply if another assumes verification side there rather still needs context unless stated directly to check it to statement confirming any basis). 

3. **One side of the model can be 1.375 inches.**
   - **Assuming correctness on a division with other integer context needing noticeable configurations given rounding would require clear dimensional cycles but it seems possible if a division noted backward of direct 5.5 requiring checks remains half potential but thus improper checks remain worth others as possible but review regarding that context.** 

4. **The scale drawing is proportional to the original.**
   - **Correct**: The pentagon and its scale drawing maintain the same shape (proportional), just differing in size due to the scale factor.

5. **The scale factor is multiplied by 5.5 to get the dimensions of the original.**
   - **Incorrect**: Actually, the dimensions of the original would be calculated by multiplying the drawn dimensions by **4** (the inverse of the one-fourth scale down concept).

### Conclusion:
Based on this analysis, the three options that apply to the original pentagon are:

1. **The scale drawing represents a reduction.**
2. **The scale drawing is proportional to the original.**
3. **The original should resize checking comparisons given depends around adaptation proportions; otherwise, checks scale base values backtrack around dimensional doubling noted otherwise along line either way to check back with core comparisons could ensure dimensional bounce saying confirms side reach specifically, old basics confirming sided; thus surrounding 22 inches long remains missing calculation dialogue thus initially requiring clarify if connected dimensions represent either segment without otherwise would reflect downside**.

Please ensure clarity on method checks in outputs ensure definitiveness around original sides as concepts checking against given shapes respectively retaining base ratio checked values would complete around requirements given data.

The scale drawing below shows an enlargement by a scale factor of 2.

A trapezoid has top side length of 12, right side length of 4.5, bottom side length of 7, and left side length of question mark. A larger trapezoid has top side length of 24, right side length of 9, bottom side length of 14, and left side length of 9.

What is the missing side length in the original trapezoid?
4.5
9
13.5
18
Mark this and return

To solve for the missing side length of the original trapezoid, we need to use the scale factor given for the enlargement.

### Given:
- The larger trapezoid has the following dimensions:
  - Top side length: 24
  - Right side length: 9
  - Bottom side length: 14
  - Left side length: 9
  
- The original trapezoid has the following dimensions:
  - Top side length: 12
  - Right side length: 4.5
  - Bottom side length: 7
  - Left side length: question mark.

### Scale Factor:
The trapezoid is enlarged by a scale factor of 2. This means that each dimension of the original trapezoid is multiplied by 2 to get the dimensions of the larger trapezoid.

### Finding the Missing Side Length:
Since we know the side lengths of the larger trapezoid, we can use the scale factor to find the original left side length. The left side length of the larger trapezoid is 9.

To find the original trapezoid's left side length, we can divide the corresponding side length of the enlarged trapezoid by the scale factor:

\[
\text{Left side length of the original trapezoid} = \frac{\text{Left side length of the enlarged trapezoid}}{\text{Scale Factor}}
\]

\[
\text{Left side length of the original trapezoid} = \frac{9}{2} = 4.5
\]

### Conclusion:
The missing side length in the original trapezoid is **4.5**.

Triangle ABC will be dilated according to the rule
D Subscript F, 0.25(x,y), where point F is the center of dilation.

On a coordinate plane, triangle A B C has points (negative 5, negative 3), (negative 5, 1), and (1, negative 1). Point F is at (3, 1).

What will be the coordinates of vertex A' of the image?

(-8,-4)
(-2,-1)
(0, 0)
(1, 0)
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To determine the coordinates of vertex \( A' \) of triangle \( ABC \) after dilation, we will use the dilation rule \( D_F, 0.25(x, y) \), where point \( F (3, 1) \) is the center of dilation and the scale factor is \( 0.25 \).

### Given Vertex A:
- The coordinates of vertex \( A \) are \( (-5, -3) \).

### Dilation Steps:
1. **Translate the coordinates relative to the center of dilation** \( F \):
   - To do this, we subtract the coordinates of point \( F \) from point \( A \):
   \[
   A_{relative} = (A_x - F_x, A_y - F_y) = (-5 - 3, -3 - 1) = (-8, -4)
   \]

2. **Apply the dilation factor of \( 0.25 \) to the relative coordinates**:
   - Now we multiply the relative coordinates by the scale factor \( 0.25 \):
   \[
   A'_{relative} = (0.25 \cdot -8, 0.25 \cdot -4) = (-2, -1)
   \]

3. **Translate back to the original coordinate system** by adding the coordinates of point \( F \):
   - Finally, we add \( F \) back to the dilated coordinates:
   \[
   A' = (A'_{relative_x} + F_x, A'_{relative_y} + F_y) = (-2 + 3, -1 + 1) = (1, 0)
   \]

### Conclusion:
The coordinates of vertex \( A' \) of the image after dilation are \( (1, 0) \).

So, the correct answer is: **(1, 0)**.

Triangle PQR was dilated according to the rule  DO,2(x,y)(2x,2y) to create similar triangle P'Q'Q.

On a coordinate plane, (0, 0) is the center of dilation. Triangle P Q R is dilated to create triangle P Q prime Q. Triangle P Q R has points (negative 2, 2), (negative 2, 4), and (negative 1, 2). Triangle P prime Q prime Q has points (negative 4, 4), (negative 4, 8), and (negative 2, 4).

Which statements are true? Select two options.

∠R corresponds to ∠P'QQ'.
∠PQR corresponds to ∠QPQ'.
Segment QQ' is parallel to segment PP'.
Side RQ corresponds to side QQ'.
△PQR ≅ △P'Q'Q
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To analyze the given triangles \( PQR \) and \( P'Q'Q \) based on their coordinates and the dilation rule, we first need to verify some information:

### Given Points:
- Triangle \( PQR \) has points:
  - \( P(-2, 2) \)
  - \( Q(-2, 4) \)
  - \( R(-1, 2) \)

- Triangle \( P'Q'Q \) has points:
  - \( P'(-4, 4) \)
  - \( Q'(-4, 8) \)
  - \( Q(-2, 4) \)

### Dilation Rule:
The dilation rule is \( DO,2(x,y) \) which means that each point is multiplied by 2 relative to the origin (0, 0) as the center of dilation.

### Analyzing the Statements:
1. **∠R corresponds to ∠P'QQ'.**  
   - **True**: In the original triangle \( PQR \), \( R \) corresponds to angle \( P'QQ' \) because vertices in the smaller triangle correspond to vertices in the larger triangle based on the same positioning.
   
2. **∠PQR corresponds to ∠QPQ'.**  
   - **False**: This statement is incorrect; \( \angle PQR \) corresponds to \( \angle P'Q'Q \) instead.

3. **Segment QQ' is parallel to segment PP'.**  
   - **True**: After the dilation, both segments are vertical lines, making them parallel.

4. **Side RQ corresponds to side QQ'.**  
   - **False**: In triangle \( PQR \), side \( RQ \) corresponds to side \( P'Q' \) in the dilated triangle, not to \( QQ' \).

5. **△PQR ≅ △P'Q'Q.**  
   - **False**: The triangles are not congruent; they are similar due to dilation, but not equal in size.

### Conclusion:
The two true statements are:
1. **∠R corresponds to ∠P'QQ'.**
2. **Segment QQ' is parallel to segment PP'.**

Therefore, those are the correct options.

Quadrilateral FGHJ is dilated according to the rule
DO,Two-thirds(x,y)(two-thirds x, two-thirds y) to create the image quadrilateral F'G'H'J', which is not shown.

On a coordinate plane, quadrilateral F G H J has points (negative 5, negative 4), (negative 3, negative 2), (negative 1, negative 4), (negative 3, negative 6).

What are the coordinates of point J'?

(-2,-4)
(-2,-6)
(Negative nine-halves, negative 4)
(Negative nine-halves, negative 9)
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To determine the coordinates of point \( J' \) after dilating quadrilateral \( FGHJ \) by the rule \( D_{O,\frac{2}{3}}(x,y) \), we need to apply the dilation factor of \( \frac{2}{3} \) to the coordinates of point \( J \).

### Step 1: Identify the Coordinates of Point \( J \)
The coordinates of point \( J \) in quadrilateral \( FGHJ \) are:
- \( J(-3, -6) \)

### Step 2: Apply the Dilation Rule
To find the dilated coordinates \( J' \), we use the dilation formula:

\[
J' = \left( \frac{2}{3} x, \frac{2}{3} y \right)
\]

Substituting the coordinates of point \( J \):

- \( x = -3 \)
- \( y = -6 \)

Calculating the coordinates of point \( J' \):

\[
J' = \left( \frac{2}{3} \cdot -3, \frac{2}{3} \cdot -6 \right)
\]

Calculating each coordinate:
1. For the x-coordinate:
   \[
   J'_x = \frac{2}{3} \cdot -3 = -2
   \]

2. For the y-coordinate:
   \[
   J'_y = \frac{2}{3} \cdot -6 = -4
   \]

Thus, the coordinates of point \( J' \) are:
\[
J' = (-2, -4)
\]

### Conclusion:
The coordinates of point \( J' \) after the dilation are **(-2, -4)**. 

Thus, the correct answer is: **(-2,-4)**.

Which graph shows a dilation?  

On a coordinate plane, 2 triangles are shown. The smaller triangle has points (negative 2, 0), (negative 2, 1), and (0, 0). The larger triangle has points (negative 3, 0), (negative 3, 4), and (2, 0).
On a coordinate plane, 2 triangles are shown. The smaller triangle has points (negative 2, 0), (negative 2, 2), and (1, 0). The larger triangle has points (negative 3, 0), (negative 3, 4), and (2, 0).
On a coordinate plane, 2 triangles are shown. The smaller triangle has points (negative 1, 0), (negative 1, 1), and (2, 0). The larger triangle has points (negative 2, 0), (negative 2, 3), and (4, 0).
On a coordinate plane, 2 triangles are shown. The smaller triangle has points (negative 1, 0), (negative 1, 2), and (3, 0). The larger triangle has points (negative 2, 0), (negative 2, 3), and (4, 0).
Mark this and return

To determine which graph shows a dilation, we need to identify which pair of triangles shows a consistent scale factor applied to all corresponding points of the smaller triangle.

### Analyzing Each Option:
1. **First Option:**
   - Smaller triangle: **(−2, 0), (−2, 1), (0, 0)** 
   - Larger triangle: **(−3, 0), (−3, 4), (2, 0)**
   
   Determine the scale factor:
   - The coordinates of the larger triangle do not maintain the same ratio in comparison to the smaller triangle's coordinates, thus it is not a dilation.

2. **Second Option:**
   - Smaller triangle: **(−2, 0), (−2, 2), (1, 0)**
   - Larger triangle: **(−3, 0), (−3, 4), (2, 0)**
   
   Determine the scale factor:
   - Ratios do not maintain a consistent scale factor across corresponding points; thus, it is not a dilation.

3. **Third Option:**
   - Smaller triangle: **(−1, 0), (−1, 1), (2, 0)**
   - Larger triangle: **(−2, 0), (−2, 3), (4, 0)**
   
   Determine the scale factor:
   - \( (-1, 0) \to (-2, 0) \) scale factor of 2 (from (-1 to -2)).
   - \( (-1, 1) \to (-2, 3) \) does not maintain consistent proportionality (vertical distance).
   - \( (2, 0) \to (4, 0) \) scale factor of 2.
   - Since all ratios are not consistent across points, it is not a dilation.

4. **Fourth Option:**
   - Smaller triangle: **(−1, 0), (−1, 2), (3, 0)**
   - Larger triangle: **(−2, 0), (−2, 3), (4, 0)**
   
   Determine the scale factor:
   - \( (-1, 0) \to (-2, 0) \) corresponds to a scale factor of 2 (double).
   - \( (-1, 2) \to (-2, 3) \) does not maintain a consistent proportionality.
   - \( (3, 0) \to (4, 0) \) corresponds to a scale factor of about \(1.33\); thus, it is inconsistent across the points.

### Conclusion:
None of the given options clearly shows a consistent scale dilation applied to all corresponding points. A true dilation would need to maintain consistent proportions across all coordinates.

However, if you return to each option and confirm there's an error in tracking points, ensuring dilation ratio determines how points correspond could clearly assess which graph might suggest proportional; thus validate accordingly to known answers specific to instructional criteria.

If you quickly review any drawn coordinate pairs, confirming ratios direct could lead you true visual inspection of a provided outline and further requirement toward exact direct measure toward control—the accurate pairing to accessing could yield a clear output based properly through standard correctly! 

If you have specific correct outputs that you'd like consideration on, please confirm measurements and point specifications as needed.

Which of these events, if added to the timeline, would come between the invention of the mercury thermometer and the lightning rod?

1711: John Shore invented the tuning fork.
1722: French Hopffer patented the fire extinguisher.
1733: John Kay invented the flying shuttle.
1757: John Campbell invented the sextant.

To determine which of the events listed would go chronologically between the invention of the mercury thermometer and the invention of the lightning rod, we must first know the dates of those inventions:

- **The mercury thermometer** was invented by **Gabriel Fahrenheit in 1714**.
- **The lightning rod** was invented by **Benjamin Franklin in 1752**.

Now, we can examine the events given:

1. **1711: John Shore invented the tuning fork.**
   - This event occurred before the mercury thermometer.

2. **1722: French Hopffer patented the fire extinguisher.**
   - This event also occurred after the mercury thermometer but before the lightning rod.

3. **1733: John Kay invented the flying shuttle.**
   - This event occurred between the invention of the mercury thermometer and the lightning rod.

4. **1757: John Campbell invented the sextant.**
   - This event occurred after the lightning rod.

### Conclusion:
The event that would come chronologically between the invention of the mercury thermometer and the lightning rod is:

**1722: French Hopffer patented the fire extinguisher.**