To solve the problem step by step, let's start with the coordinates of triangle \( ABC \):
- A(5, -1)
- B(5, -9)
- C(2, -7)
Question 1: Rotate \( \triangle ABC \) 90 degrees counterclockwise about the origin to create \( \triangle A'B'C' \).
For a 90-degree counterclockwise rotation about the origin, the transformation of a point \((x, y)\) to \((x', y')\) is given by: \[ (x', y') = (-y, x) \]
Applying this transformation to each point:
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For A(5, -1): \[ A' = (-(-1), 5) = (1, 5) \]
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For B(5, -9): \[ B' = (-(-9), 5) = (9, 5) \]
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For C(2, -7): \[ C' = (-(-7), 2) = (7, 2) \]
So the coordinates for \( \triangle A'B'C' \) are:
- A' (1, 5)
- B' (9, 5)
- C' (7, 2)
Question 2: Reflect \( \triangle A'B'C' \) across the y-axis to create \( \triangle A"B"C" \).
For reflection across the y-axis, the transformation is given by: \[ (x', y') = (-x, y) \]
Applying this transformation to each point:
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For A' (1, 5): \[ A" = (-1, 5) \]
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For B' (9, 5): \[ B" = (-9, 5) \]
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For C' (7, 2): \[ C" = (-7, 2) \]
So the coordinates for \( \triangle A"B"C" \) are:
- A" (-1, 5)
- B" (-9, 5)
- C" (-7, 2)
Question 3: True or False? \( \triangle ABC \cong \triangle A"B"C" \)
To determine if the triangles are congruent, we need to check if all corresponding sides are equal in length.
The original triangle \( ABC \):
- A(5, -1) to B(5, -9): \[ |y_2 - y_1| = |-9 - (-1)| = 8 \]
- B(5, -9) to C(2, -7): \[ \sqrt{(2-5)^2 + (-7 + 9)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
- C(2, -7) to A(5, -1): \[ \sqrt{(5 - 2)^2 + (-1 + 7)^2} = \sqrt{(3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45} \]
The lengths of the sides of \( \triangle ABC \) are \( 8, \sqrt{13}, \sqrt{45} \).
Now for \( \triangle A"B"C" \):
- A"(-1, 5) to B"(-9, 5): \[ |x_2 - x_1| = |-9 - (-1)| = 8 \]
- B"(-9, 5) to C"(-7, 2): \[ \sqrt{(-7 + 9)^2 + (2 - 5)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
- C"(-7, 2) to A"(-1, 5): \[ \sqrt{(-1 + 7)^2 + (5 - 2)^2} = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} \]
The lengths of the sides of \( \triangle A"B"C" \) are also \( 8, \sqrt{13}, \sqrt{45} \).
Since all corresponding sides are equal, the triangles are congruent.
Final responses:
- A' (1, 5), B' (9, 5), C' (7, 2)
- A" (-1, 5), B" (-9, 5), C" (-7, 2)
- True
1 answer