Question
Which sequence of translations, rotations, and reflections produces △P’Q’R’ from the pre-image △PQR?
On a coordinate plane, triangle P Q R has points (2.5, 1), (0, negative 3), (5, negative 1). Triangle P prime Q prime R prime has points (5, 2.5), (3, negative 1), (1, 6).
a reflection across the y-axis followed by a clockwise rotation 90° about the origin
a reflection across the x-axis, followed by a counter-clockwise rotation 90° about the origin, followed by a translation 4 units down and 4 units to the left
a counter-clockwise rotation of 90° about the origin followed by a translation 4 units to the right
no sequence of translations, rotations, and reflections because the images are not congruent
On a coordinate plane, triangle P Q R has points (2.5, 1), (0, negative 3), (5, negative 1). Triangle P prime Q prime R prime has points (5, 2.5), (3, negative 1), (1, 6).
a reflection across the y-axis followed by a clockwise rotation 90° about the origin
a reflection across the x-axis, followed by a counter-clockwise rotation 90° about the origin, followed by a translation 4 units down and 4 units to the left
a counter-clockwise rotation of 90° about the origin followed by a translation 4 units to the right
no sequence of translations, rotations, and reflections because the images are not congruent
Answers
GPT-4o mini
To determine which sequence of transformations maps triangle \( PQR \) to triangle \( P'Q'R' \), let's first analyze the coordinates of the triangles.
The coordinates of triangle \( PQR \) are:
- \( P(2.5, 1) \)
- \( Q(0, -3) \)
- \( R(5, -1) \)
The coordinates of triangle \( P'Q'R' \) are:
- \( P'(5, 2.5) \)
- \( Q'(3, -1) \)
- \( R'(1, 6) \)
Next, let's compare the transformations suggested in the options.
1. **Check for congruence**:
- The distance between corresponding points in \( PQR \) and \( P'Q'R' \) should be the same for the triangles to be congruent.
- Calculate the lengths of the sides of both triangles using the distance formula:
- For \( PQR \):
- \( PQ: \sqrt{(2.5 - 0)^2 + (1 + 3)^2} = \sqrt{(2.5)^2 + (4)^2} = \sqrt{6.25 + 16} = \sqrt{22.25} \)
- \( QR: \sqrt{(0 - 5)^2 + (-3 + 1)^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \)
- \( RP: \sqrt{(5 - 2.5)^2 + (-1 - 1)^2} = \sqrt{(2.5)^2 + (-2)^2} = \sqrt{6.25 + 4} = \sqrt{10.25} \)
- For \( P'Q'R' \):
- \( P'Q': \sqrt{(5 - 3)^2 + (2.5 + 1)^2} = \sqrt{(2)^2 + (3.5)^2} = \sqrt{4 + 12.25} = \sqrt{16.25} \)
- \( Q'R': \sqrt{(3 - 1)^2 + (-1 - 6)^2} = \sqrt{(2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \)
- \( R'P': \sqrt{(1 - 5)^2 + (6 - 2.5)^2} = \sqrt{(-4)^2 + (3.5)^2} = \sqrt{16 + 12.25} = \sqrt{28.25} \)
Since the corresponding side lengths are not equal, we can conclude that the triangles are not congruent. This means that the last option—**no sequence of translations, rotations, and reflections because the images are not congruent**—is the correct choice.
Thus, the answer is:
**no sequence of translations, rotations, and reflections because the images are not congruent.**
The coordinates of triangle \( PQR \) are:
- \( P(2.5, 1) \)
- \( Q(0, -3) \)
- \( R(5, -1) \)
The coordinates of triangle \( P'Q'R' \) are:
- \( P'(5, 2.5) \)
- \( Q'(3, -1) \)
- \( R'(1, 6) \)
Next, let's compare the transformations suggested in the options.
1. **Check for congruence**:
- The distance between corresponding points in \( PQR \) and \( P'Q'R' \) should be the same for the triangles to be congruent.
- Calculate the lengths of the sides of both triangles using the distance formula:
- For \( PQR \):
- \( PQ: \sqrt{(2.5 - 0)^2 + (1 + 3)^2} = \sqrt{(2.5)^2 + (4)^2} = \sqrt{6.25 + 16} = \sqrt{22.25} \)
- \( QR: \sqrt{(0 - 5)^2 + (-3 + 1)^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \)
- \( RP: \sqrt{(5 - 2.5)^2 + (-1 - 1)^2} = \sqrt{(2.5)^2 + (-2)^2} = \sqrt{6.25 + 4} = \sqrt{10.25} \)
- For \( P'Q'R' \):
- \( P'Q': \sqrt{(5 - 3)^2 + (2.5 + 1)^2} = \sqrt{(2)^2 + (3.5)^2} = \sqrt{4 + 12.25} = \sqrt{16.25} \)
- \( Q'R': \sqrt{(3 - 1)^2 + (-1 - 6)^2} = \sqrt{(2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \)
- \( R'P': \sqrt{(1 - 5)^2 + (6 - 2.5)^2} = \sqrt{(-4)^2 + (3.5)^2} = \sqrt{16 + 12.25} = \sqrt{28.25} \)
Since the corresponding side lengths are not equal, we can conclude that the triangles are not congruent. This means that the last option—**no sequence of translations, rotations, and reflections because the images are not congruent**—is the correct choice.
Thus, the answer is:
**no sequence of translations, rotations, and reflections because the images are not congruent.**