Question
a dilation has a center of 0,0. Find the vertices of the line segment for the given scale factor.
A (-3, 0)
B (-2, 3)
SF: 3
A' ?,?
B' ?, ?
A (-3, 0)
B (-2, 3)
SF: 3
A' ?,?
B' ?, ?
Answers
GPT-4o mini
A dilation with a center at the origin (0, 0) stretches or shrinks points relative to that center by a specified scale factor (SF). For a scale factor of 3, each coordinate of the points is multiplied by 3.
Given the points:
- A(-3, 0)
- B(-2, 3)
We apply the dilation:
For point A:
\[
A' = (x_A \cdot SF, y_A \cdot SF) = (-3 \cdot 3, 0 \cdot 3) = (-9, 0)
\]
For point B:
\[
B' = (x_B \cdot SF, y_B \cdot SF) = (-2 \cdot 3, 3 \cdot 3) = (-6, 9)
\]
So the vertices of the dilated line segment are:
- A' (-9, 0)
- B' (-6, 9)
Given the points:
- A(-3, 0)
- B(-2, 3)
We apply the dilation:
For point A:
\[
A' = (x_A \cdot SF, y_A \cdot SF) = (-3 \cdot 3, 0 \cdot 3) = (-9, 0)
\]
For point B:
\[
B' = (x_B \cdot SF, y_B \cdot SF) = (-2 \cdot 3, 3 \cdot 3) = (-6, 9)
\]
So the vertices of the dilated line segment are:
- A' (-9, 0)
- B' (-6, 9)
a dilation has a center of 2,1. Find the vertices of the line segment for the given scale factor.
D(0,3)
E(4,3)
SF: 1/2
D' ?,?
E' ?,?
D(0,3)
E(4,3)
SF: 1/2
D' ?,?
E' ?,?
GPT-4o mini
To find the vertices of the dilated line segment with respect to a center of dilation (2, 1) and a scale factor (SF) of \( \frac{1}{2} \), we need to perform the following steps:
1. **Translate the points so that the center of dilation becomes the origin.**
This involves subtracting the coordinates of the center of dilation (2, 1) from each point.
2. **Apply the scale factor to the translated points.**
3. **Translate the points back to their original position by adding the center of dilation (2, 1) back to them.**
### Step 1: Translate the points
- For point D(0, 3):
\[
D_{translated} = (0 - 2, 3 - 1) = (-2, 2)
\]
- For point E(4, 3):
\[
E_{translated} = (4 - 2, 3 - 1) = (2, 2)
\]
### Step 2: Apply the scale factor \( \frac{1}{2} \)
- For point D:
\[
D'_{scaled} = \left(-2 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}\right) = (-1, 1)
\]
- For point E:
\[
E'_{scaled} = \left(2 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}\right) = (1, 1)
\]
### Step 3: Translate back to the original position
- For point D':
\[
D' = (-1 + 2, 1 + 1) = (1, 2)
\]
- For point E':
\[
E' = (1 + 2, 1 + 1) = (3, 2)
\]
### Final Results
So the vertices of the dilated line segment are:
- \( D' (1, 2) \)
- \( E' (3, 2) \)
1. **Translate the points so that the center of dilation becomes the origin.**
This involves subtracting the coordinates of the center of dilation (2, 1) from each point.
2. **Apply the scale factor to the translated points.**
3. **Translate the points back to their original position by adding the center of dilation (2, 1) back to them.**
### Step 1: Translate the points
- For point D(0, 3):
\[
D_{translated} = (0 - 2, 3 - 1) = (-2, 2)
\]
- For point E(4, 3):
\[
E_{translated} = (4 - 2, 3 - 1) = (2, 2)
\]
### Step 2: Apply the scale factor \( \frac{1}{2} \)
- For point D:
\[
D'_{scaled} = \left(-2 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}\right) = (-1, 1)
\]
- For point E:
\[
E'_{scaled} = \left(2 \cdot \frac{1}{2}, 2 \cdot \frac{1}{2}\right) = (1, 1)
\]
### Step 3: Translate back to the original position
- For point D':
\[
D' = (-1 + 2, 1 + 1) = (1, 2)
\]
- For point E':
\[
E' = (1 + 2, 1 + 1) = (3, 2)
\]
### Final Results
So the vertices of the dilated line segment are:
- \( D' (1, 2) \)
- \( E' (3, 2) \)
If two corresponding interior angles of two triangles are congruent, which theorem would be used to know the triangles are similar?
GPT-4o mini
If two corresponding interior angles of two triangles are congruent, you can use the **Angle-Angle (AA) Similarity Postulate** to conclude that the triangles are similar. The AA Similarity Postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. This method works because the third angle in each triangle must also be congruent, which satisfies the criteria for triangle similarity.