Let's address each question based on the information provided:
Question 1
To dilate triangle ABC with respect to the origin by a magnitude of 3, you multiply each coordinate of the vertices of triangle ABC by 3.
If the original coordinates are A(x1, y1), B(x2, y2), and C(x3, y3), the coordinates after dilation will be:
- A′(3 * x1, 3 * y1)
- B′(3 * x2, 3 * y2)
- C′(3 * x3, 3 * y3)
Assuming the coordinates of A, B, and C are not provided but are relevant to the possible answer choices, we choose the set of points suggested (without specific coordinates in this case).
Question 2
Magnitude of dilation can be determined from the original and the new coordinates of two corresponding points, say A and A′.
- If the coordinates of point A are (x1, y1) and the coordinates of point A' are (x1', y1'), then the magnitude of dilation is the ratio of the distance from the center of dilation to A, and the distance from the center of dilation to A′.
The options suggest possible magnitudes of dilation. Without specific coordinates in this question, choose based on stipulations of diminishing or enlarging area—hence you would evaluate the change from polygon ABCD to A′B′C′D′.
Question 3
Same logic applies as in Question 2 for determining the magnitude of dilation in parallelogram ABCD to A′B′C′D′. The dilation factor will be either greater than or less than 1 based on the size comparison of corresponding sides.
Question 4
To dilate triangle XYZ:
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The center of dilation is (4, 7).
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The formula for dilation using a center (x0, y0) and scale factor k is: \[ X' = x0 + k(x - x0) \] \[ Y' = y0 + k(y - y0) \]
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For point X(0, 4):
- X′ = 4 + 12(0 - 4) = 4 - 48 = -44 (but this option is not in constraints)
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Proceed similarly for Y and Z and match relation to find correct scaling points.
Question 5
To find the new coordinates of point A after dilation around the point (-2, -1) with a scale factor of 2:
- Start with A(x, y).
- You apply: \[ A' = (-2) + 2(A_x + 2) \] \[ B' = (-1) + 2(A_y + 1) \] If original A had (x,y) (assuming from any selected answer point).
You’ll compute that point based on provided shifting defined through magnitude.
In preparation for answering these questions, specific coordinates for points A, B, C, D were assumed for numerical abstraction. The precise numbers should ideally be provided to compute exact outcomes per triangle/polygon specifics accordingly.
1 answer