To find the coordinates of the transformed points through the described transformations, we'll first perform a 90-degree counterclockwise rotation about the origin, and then reflect the resulting points across the y-axis.
Question 1: Rotate △ABC 90 degrees counterclockwise.
The formula for rotating a point \((x, y)\) 90 degrees counterclockwise around the origin is: \[ (x, y) \rightarrow (-y, x) \]
Now, applying this to the original points:
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For point A \((5, -1)\): \[ A' = (-(-1), 5) = (1, 5) \]
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For point B \((5, -9)\): \[ B' = (-(-9), 5) = (9, 5) \]
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For point C \((2, -7)\): \[ C' = (-(-7), 2) = (7, 2) \]
Therefore, the coordinates after rotation are:
- A′ \( (1, 5) \)
- B′ \( (9, 5) \)
- C′ \( (7, 2) \)
Question 2: Reflect △A′B′C′ across the y-axis.
The formula for reflecting a point \((x, y)\) across the y-axis is: \[ (x, y) \rightarrow (-x, y) \]
Now applying reflection to the rotated points:
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For point A′ \( (1, 5) \): \[ A" = (-1, 5) \]
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For point B′ \( (9, 5) \): \[ B" = (-9, 5) \]
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For point C′ \( (7, 2) \): \[ C" = (-7, 2) \]
Therefore, the coordinates after reflection are:
- A" \( (-1, 5) \)
- B" \( (-9, 5) \)
- C" \( (-7, 2) \)
Question 3: True or False? △ABC ≅ △A"B"C"
To determine if triangles \(△ABC\) and \(△A"B"C"\) are congruent, we check if the corresponding sides have the same lengths.
The lengths of the sides of triangle \(△ABC\) can be found using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating the lengths of \(△ABC\):
- Length AB: \[ \sqrt{(5 - 5)^2 + (-1 + 9)^2} = \sqrt{0 + 8^2} = 8 \]
- Length BC: \[ \sqrt{(5 - 2)^2 + (-9 + 7)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
- Length CA: \[ \sqrt{(2 - 5)^2 + (-7 + 1)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \]
Now calculating the lengths of \(△A"B"C"\):
- Length A"B": \[ \sqrt{(-1 - (-9))^2 + (5 - 5)^2} = \sqrt{8^2 + 0} = 8 \]
- Length B"C": \[ \sqrt{(-9 - (-7))^2 + (5 - 2)^2} = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
- Length C"A: \[ \sqrt{(-7 - (-1))^2 + (2 - 5)^2} = \sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]
Since the lengths of the corresponding sides of the two triangles are equal:
- AB = A"B' (8)
- BC = B"C' (\(\sqrt{13}\))
- CA = C"A (3\(\sqrt{5}\))
Thus, we conclude that △ABC ≅ △A"B"C".
Final response for Question 3: True.