Transformations and Symmetry
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ISEE Middle Level: Quantitative Reasoning › Transformations and Symmetry
A new shape is created by joining two congruent right isosceles triangles along their longest sides (hypotenuses). Which statement best describes the symmetry of the resulting quadrilateral?
It has two lines of symmetry and rotational symmetry of order 2.
It has four lines of symmetry and rotational symmetry of order 4.
It has one line of symmetry and no rotational symmetry.
It has no lines of symmetry but has rotational symmetry of order 2.
Explanation
When two congruent right isosceles triangles are joined along their hypotenuses, they form a square. A square has four lines of symmetry (two through the midpoints of opposite sides, and two through opposite vertices). It also has rotational symmetry of order 4, as it maps onto itself after rotations of 90, 180, and 270 degrees.
A certain polygon is known to have at least one pair of parallel sides. It is also known to have rotational symmetry of 180 degrees. Based on this information, which of the following must be true about the polygon?
The polygon must have two pairs of parallel sides.
The polygon must have four equal angles.
The polygon must have at least two lines of symmetry.
The polygon must be a regular polygon.
Explanation
A polygon with 180-degree rotational symmetry must have opposite sides that are both parallel and equal in length. Since the polygon is already given to have at least one pair of parallel sides, the rotational symmetry guarantees that the other pair of opposite sides must also be parallel. A quadrilateral with two pairs of parallel sides is a parallelogram. Therefore, the polygon must have two pairs of parallel sides.
A right triangle has vertices at A(2, 1), B(5, 1), and C(2, 5). The triangle is reflected across the y-axis to create a new triangle, A'B'C'. Which of the following statements about triangle A'B'C' is true?
Triangle A'B'C' is congruent to triangle ABC.
Triangle A'B'C' has at least one line of symmetry.
Triangle A'B'C' has a vertex at the coordinate (5, –1).
The area of triangle A'B'C' is different from the area of triangle ABC.
Explanation
A reflection is a rigid transformation, which means it preserves size, shape, side lengths, and angle measures. Therefore, the resulting triangle A'B'C' is congruent to the original triangle ABC. Area is preserved. A reflection across the y-axis changes (x, y) to (–x, y), so no vertex will be at (5, –1). The original triangle is a scalene right triangle, so neither it nor its reflection has a line of symmetry.
A regular nonagon is a nine-sided polygon with all sides and angles equal. What is the smallest positive angle of rotation that will map the nonagon onto itself?
40 degrees
90 degrees
9 degrees
20 degrees
Explanation
For a regular polygon with n sides, the smallest angle of rotation that maps it onto itself is found by dividing 360 degrees by the number of sides, n. For a regular nonagon (n=9), the angle is \(360^\circ / 9 = 40^\circ\).
A point P has coordinates (–3, 5). It is first reflected across the y-axis to create point P'. Then, P' is translated 4 units down and 2 units to the left to create point P''. What are the coordinates of P''?
(5, 9)
(–1, –1)
(–5, –9)
(1, 1)
Explanation
First, reflecting P(–3, 5) across the y-axis changes the sign of the x-coordinate, resulting in P'(3, 5). Next, translating P'(3, 5) 4 units down and 2 units to the left means subtracting 4 from the y-coordinate and 2 from the x-coordinate: (3 – 2, 5 – 4), which results in P''(1, 1).
A regular decagon is a polygon with ten equal sides and ten equal interior angles. How many lines of symmetry does a regular decagon possess?
2
5
10
20
Explanation
A regular polygon with n sides has n lines of symmetry. Since a regular decagon has 10 sides, it has 10 lines of symmetry. These lines pass through opposite vertices or the midpoints of opposite sides.
A transformation on the coordinate plane maps every point (x, y) to the point (–y, x). This transformation is equivalent to which of the following?
A reflection across the y-axis followed by a reflection across the x-axis
A 90-degree counterclockwise rotation about the origin
A reflection across the line y = –x
A 90-degree clockwise rotation about the origin
Explanation
Let's test a point, for example (2, 3). The transformation maps it to (–3, 2). This corresponds to a 90-degree counterclockwise rotation about the origin. The rule for a 90-degree counterclockwise rotation is (x, y) → (–y, x).
A reflection in the coordinate plane transforms point A(7, 2) to point A'(–1, 2). What is the equation of the line of reflection?
y = 2
x = 3
y = x
x = 4
Explanation
Since the y-coordinate does not change, the line of reflection must be a vertical line of the form x = k. A line of reflection is the perpendicular bisector of the segment connecting a point and its image. The midpoint of the segment AA' has an x-coordinate that is the average of the x-coordinates of A and A': (7 + (–1)) / 2 = 6 / 2 = 3. Thus, the line of reflection is x = 3.
A point (a, b) in the coordinate plane is reflected across the x-axis. The resulting point is then reflected across the y-axis. This sequence of two reflections is equivalent to which single transformation?
A 90-degree counterclockwise rotation about the origin
A reflection across the line y = –x
A 180-degree rotation about the origin
A reflection across the line y = x
Explanation
Reflecting the point (a, b) across the x-axis gives (a, –b). Reflecting this new point (a, –b) across the y-axis gives (–a, –b). The transformation that takes (a, b) to (–a, –b) is a 180-degree rotation about the origin.
A point at (4, –2) is reflected across the line y = x. The resulting point is then rotated 90 degrees clockwise about the origin. What are the final coordinates of the point?
(2, –4)
(–2, 4)
(4, 2)
(–4, –2)
Explanation
First, reflect the point (4, –2) across the line y = x. The rule for this reflection is (x, y) → (y, x), so the point becomes (–2, 4). Next, rotate this new point 90 degrees clockwise about the origin. The rule for this rotation is (x, y) → (y, –x). Applying this to (–2, 4), we get (4, –(–2)), which simplifies to (4, 2).