Question
Which graph shows the result of dilating this figure by a factor of 4 about the origin?
On a coordinate plane, rectangle A B C D has points (negative 1, 1), (3, 1), (3, negative 1), (negative 1, negative 1).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 4, 4), (12, 4), (12, negative 4), (negative 4, negative 4).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 2, 2), (6, 2), (6, negative 2), (negative 2, negative 2).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 0.25, 0.25), (0.75, 0.25), (0.75, negative 0.25). (negative 0.25, negative 0.25).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 0.5, 0.5), (1.5, 0.5), (1.5, negative 0.5), (negative 0.5, negative 0.5).
On a coordinate plane, rectangle A B C D has points (negative 1, 1), (3, 1), (3, negative 1), (negative 1, negative 1).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 4, 4), (12, 4), (12, negative 4), (negative 4, negative 4).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 2, 2), (6, 2), (6, negative 2), (negative 2, negative 2).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 0.25, 0.25), (0.75, 0.25), (0.75, negative 0.25). (negative 0.25, negative 0.25).
On a coordinate plane, rectangle A prime B prime C prime D prime has points (negative 0.5, 0.5), (1.5, 0.5), (1.5, negative 0.5), (negative 0.5, negative 0.5).
Answers
GPT-4o mini
To find the result of dilating the rectangle defined by the points A(-1, 1), B(3, 1), C(3, -1), and D(-1, -1) by a factor of 4 about the origin, we need to multiply each coordinate of the original points by the factor of 4.
The original points are:
- A(-1, 1)
- B(3, 1)
- C(3, -1)
- D(-1, -1)
Now let's apply the dilation:
1. For point A:
\((-1, 1) \rightarrow (-1 \times 4, 1 \times 4) = (-4, 4)\)
2. For point B:
\((3, 1) \rightarrow (3 \times 4, 1 \times 4) = (12, 4)\)
3. For point C:
\((3, -1) \rightarrow (3 \times 4, -1 \times 4) = (12, -4)\)
4. For point D:
\((-1, -1) \rightarrow (-1 \times 4, -1 \times 4) = (-4, -4)\)
So after the dilation, the new coordinates of the rectangle are:
- A'(-4, 4)
- B'(12, 4)
- C'(12, -4)
- D'(-4, -4)
Now let's find which of the provided options matches these new coordinates.
The correct dilated rectangle has the points:
- (−4, 4)
- (12, 4)
- (12, −4)
- (−4, −4)
This matches the first option:
> On a coordinate plane, rectangle A' B' C' D' has points (negative 4, 4), (12, 4), (12, negative 4), (negative 4, negative 4).
Therefore, the graph that shows the result of dilating the figure by a factor of 4 about the origin is the first option.
The original points are:
- A(-1, 1)
- B(3, 1)
- C(3, -1)
- D(-1, -1)
Now let's apply the dilation:
1. For point A:
\((-1, 1) \rightarrow (-1 \times 4, 1 \times 4) = (-4, 4)\)
2. For point B:
\((3, 1) \rightarrow (3 \times 4, 1 \times 4) = (12, 4)\)
3. For point C:
\((3, -1) \rightarrow (3 \times 4, -1 \times 4) = (12, -4)\)
4. For point D:
\((-1, -1) \rightarrow (-1 \times 4, -1 \times 4) = (-4, -4)\)
So after the dilation, the new coordinates of the rectangle are:
- A'(-4, 4)
- B'(12, 4)
- C'(12, -4)
- D'(-4, -4)
Now let's find which of the provided options matches these new coordinates.
The correct dilated rectangle has the points:
- (−4, 4)
- (12, 4)
- (12, −4)
- (−4, −4)
This matches the first option:
> On a coordinate plane, rectangle A' B' C' D' has points (negative 4, 4), (12, 4), (12, negative 4), (negative 4, negative 4).
Therefore, the graph that shows the result of dilating the figure by a factor of 4 about the origin is the first option.