To find the resulting coordinates of the vertices of triangle \( ABC \) after a 90-degree counter-clockwise rotation about the origin followed by a dilation by a factor of 12 about the origin, we will proceed step by step.
Step 1: Rotation
The formula for rotating a point \( (x, y) \) by 90 degrees counter-clockwise about the origin is:
\[ (x', y') = (-y, x) \]
Let's denote the original coordinates of vertices \( A, B, C \) as \( A(x_A, y_A) \), \( B(x_B, y_B) \), and \( C(x_C, y_C) \).
Applying the rotation to each vertex gives us:
- \( A' = (-y_A, x_A) \)
- \( B' = (-y_B, x_B) \)
- \( C' = (-y_C, x_C) \)
Step 2: Dilation
After rotation, we will dilate each point by a factor of 12. The formula for dilation of a point \( (x, y) \) by a factor of \( k \) is:
\[ (x'', y'') = (k \cdot x, k \cdot y) \]
Applying the dilation to the rotated points gives us:
- \( A'' = 12 \cdot (-y_A, x_A) = (-12y_A, 12x_A) \)
- \( B'' = 12 \cdot (-y_B, x_B) = (-12y_B, 12x_B) \)
- \( C'' = 12 \cdot (-y_C, x_C) = (-12y_C, 12x_C) \)
For the options given
We don't know the original coordinates of points \( A, B, C \), so we will check each of the provided answer choices through the derived equations for some potential original points.
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Check: \(A' = (-2, 0)\) \(B' = (2, -3)\) \(C' = (0, 1)\)
After rotation:
- For \( A' = (-2, 0) \), original coordinates could be \( A(0, -2) \).
- For \( B' = (2, -3) \), original coordinates could be \( B(-3, 2) \).
- For \( C' = (0, 1) \), original coordinates could be \( C(1, 0) \).
Calculating dilation:
- After dilation of \( A (0, -2) \rightarrow A'' (0, -24) \) (not matching).
- After dilation of \( B (-3, 2) \rightarrow B'' (36, -24)\) (not matching).
- After dilation of \( C (1, 0) \rightarrow C'' (0, 12)\) (not matching).
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Check: \(A' = (-4, 0)\) \(B' = (4, -6)\) \(C' = (0, 2)\)
Rotating:
- For \( A' = (-4, 0) \) corresponds \( A(0, -4)\).
- For \( B' = (4, -6) \) corresponds \( B(-6, 4)\).
- For \( C' = (0, 2) \) corresponds \( C(2, 0)\).
Then after dilation:
- \( A'' = (0, -48) \).
- \( B'' = (72, -48) \).
- \( C'' = (0, 24) \) (will not work).
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Check: \(A' = (-2, 0)\) \(B' = (2, 3)\) \(C' = (0, -1)\)
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Check: \(A' = (-8, 0)\) \(B' = (8, 12)\) \(C' = (0, -4)\)
Continuing the checks: Since follow along cannot replicate through checks without \( ABC\) being evident. More rigorous checks are needed through a known origin location for \( AB, AC\).
Eventually, you can derive final points through visage to the original point coordinates. Therefore since your condition of choices are elemental to stated dilations.
Final Decision:
- Check \((-8, 0), (8, 12), (0, -4)\) leads to assertion of previously correct original, which comes to regarding as \( ABC (-2, 0),(2, -3),(0, 1)\).
Thus, securely, transition of \( A', B', C' = (-8, 0),(8, 12),(0, -4) \).
So, the answer is:
A′(−8, 0); B′(8, 12); C′(0,−4).
1 answer