Symmetry of polygons: lines of symmetry and rotations that map a shape onto itself
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
Ari reflects a figure across x = 1, then across x = 4. What single motion does the same, and why?
- A slide 6 units right, twice the distance between the lines
- A slide 6 units left, twice the distance between the two lines
- A slide 3 units right, the distance between the lines
- A 180° turn about (2.5, 0), the point between the lines
Read this first
Two mirror lines meet at the origin at a 45° angle
The x-axis and the line y = x. A figure is reflected across the x-axis first, then across y = x.
Bo reflects a figure across the x-axis, then across y = x. What single motion does the same, with its center and angle?
- A 45° counterclockwise turn about (0, 0), the same as the angle between the lines
- A 90° clockwise turn about (0, 0), twice the 45° between the lines
- A 90° counterclockwise turn about (0, 0), twice the 45° between the lines
- A reflection across the y-axis, which swaps left and right
A figure is reflected across y = 2 first, then across y = x + 2. Esme reflects a figure across y = 2, then across y = x + 2. What single motion does the same, with its center and angle?
- A 90° clockwise turn about (0, 2), twice the 45° between the lines
- A 90° counterclockwise turn about (0, 2), twice the 45° between the lines
- A 45° counterclockwise turn about (0, 2), the same as the angle between the lines
- A 90° counterclockwise turn about (0, 0), the origin
Jun reflects a figure across x = −1, then across y = 3. What single motion does the same, with its center and angle?
- A 180° turn about (−1, 3), twice the 90° between the lines
- A 180° turn about (0, 0), the origin
- A slide 2 left and 6 up, twice each coordinate
- A 90° turn about (−1, 3), which is the same as the angle between the lines
Kira says the square with corners (1, 1), (5, 1), (5, 5) and (1, 5) has more symmetries than a rectangle. Which list is complete?
- Four reflections (x = 3, y = 3, y = x, y = −x + 6) and turns of 90°, 180° and 270° about (0, 0)
- Four reflections (x = 3, y = 3, y = x, y = −x + 6) and a 180° turn about (3, 3) alone
- Two reflections (x = 3 and y = 3) and turns of 90°, 180° and 270° about (3, 3)
- Four reflections (x = 3, y = 3, y = x, y = −x + 6) and turns of 90°, 180° and 270° about (3, 3)
Luz draws a house shape on a grid. Which list gives all of its symmetries?
- No symmetry: the five sides of the house differ in length
- Reflections across x = 2 and y = 2.5, one vertical and one horizontal
- One reflection, across x = 2; no turn maps the house onto itself
- One reflection, across x = 2, and a 180° turn about (2, 2.5)
Milo plots a right triangle with legs 4 and 4. Which list gives all of its symmetries?
- One reflection, across y = x, and a 180° turn about (2, 2)
- Reflections across y = x and the x-axis, and a 90° turn about (0, 0)
- One reflection, across y = x, and no turns
- No symmetry, because a right triangle has no fold
Noah draws an equilateral triangle with corners (0, 0), (6, 0) and (3, 3√3). Which list gives all of its symmetries?
- Three reflections, each through one corner and the opposite midpoint, and a 180° turn about (3, √3)
- Three reflections, each through one corner and the opposite midpoint, and turns of 120° and 240° about (3, 0)
- Three reflections, each through a corner and the opposite midpoint, and turns of 120° and 240° about (3, √3)
- One reflection, across x = 3, and no turns, since just the line through the top corner folds it
Proof
1. Let ℓ be the bisector of ∠A. 2. The reflection across ℓ maps ray AB onto ray AC. 3. ? 4. So the reflection maps ∠B onto ∠C. 5. Reflections keep angle measure, so ∠B = ∠C.
Pia's proof that the base angles of isosceles △ABC (AB = AC) are equal has a gap in step 3. Which statement belongs there?
- AB = AC, so B is as far from A on the ray as C is, and the reflection puts B on C
- A is the vertex, so A and C swap places under the reflection across ℓ
- Since ∠B = ∠C, the reflection must send B to C, because equal angles fix the points
- BC maps onto itself, so B and C each stay fixed in place under the reflection across ℓ
Ren claims: every quadrilateral with a line of symmetry has a pair of equal opposite sides. Which figure breaks the claim?
- A rhombus: it has mirror lines, and the four sides are equal in length
- An isosceles trapezoid: it has a mirror line, and its two slanted sides are equal
- A kite with sides 3, 3, 5, 5 in order: a mirror line, but no equal opposite sides
- A rectangle: it has mirror lines, and both pairs of opposite sides are equal
