Rigid Motions and Transformations: math practice, Grade 10 – download the PDF

1 Translating points ★★★

Translate each point by the vector \(\langle -4, 5\rangle\).

1. \(P(3, -2)\)
2. \(Q(0, 6)\)

2 Three reflections ★★★

1. Reflect \(A(4, 7)\) in the x-axis.
2. Reflect \(B(-3, 2)\) in the y-axis.
3. Reflect \(C(5, 5)\) in the line \(y = x\). What do you notice?

3 Turning about the origin ★★★

Rotate about the origin: (a) \(A(3, 1)\) by \(90^\circ\) counterclockwise; (b) \(B(-2, 4)\) by \(180^\circ\); (c) \(C(5, -2)\) by \(270^\circ\) counterclockwise.

4 True or false? ★★★

Decide whether each statement is true or false and justify.

1. A translation can make a figure bigger.
2. A reflection reverses the orientation of a figure.
3. A rotation of \(360^\circ\) maps every figure onto itself.
4. A dilation with scale factor 2 is a rigid motion.

5 Counting symmetries ★★★

For each figure, give the number of lines of symmetry and the smallest angle of rotational symmetry (write “none” if only \(360^\circ\) works): (a) a rectangle that is not a square; (b) an equilateral triangle; (c) a regular hexagon; (d) an isosceles trapezoid; (e) a parallelogram that is not a rectangle or a rhombus.

6 Finding the translation ★★★

A translation maps \(A(2, 5)\) to \(A′(-1, 9)\).

1. Find its vector.
2. Find the image of \(B(0, 0)\).
3. Find the preimage of \(C′(7, -2)\).

7 Lengths do not change ★★★

Let \(P(1, 2)\) and \(Q(5, 5)\). Translate the segment by \(\langle 3, -4\rangle\). Compute \(PQ\) and \(P′Q′\) and compare.

8 A vertical mirror line ★★★

The line of reflection is \(x = 3\). Find the images of \((7, 2)\), \((1, -4)\) and \((3, 6)\), then write the rule for any point \((x, y)\).

9 A horizontal mirror line ★★★

The line of reflection is \(y = -1\). Show that the rule is \((x, y)\to(x, -2 - y)\), then reflect \((2, 3)\), \((-4, -1)\) and \((0, -6)\).

10 Reflecting a triangle in y = x ★★★

Triangle \(ABC\) has \(A(1, 4)\), \(B(3, 0)\), \(C(5, 2)\). Reflect it in the line \(y = x\) and compare the areas of the two triangles.

11 Order matters ★★★

Let \(T\) be the translation by \(\langle 2, 3\rangle\) and \(M\) the reflection in the y-axis. Find the image of \(P(4, -1)\) when you (a) apply \(T\) then \(M\); (b) apply \(M\) then \(T\). Are the results equal?

12 A clockwise quarter turn ★★★

Rotate \(D(2, 0)\), \(E(4, 1)\), \(F(3, 3)\) by \(90^\circ\) clockwise about the origin. Which counterclockwise angle gives the same image?

13 Drone on a grid map ★★★

A drone starts at \((2, 3)\) on a grid map (units: meters). It makes three moves, which are translations: \(\langle 4, 1\rangle\), then \(\langle -6, 2\rangle\), then \(\langle 1, -7\rangle\).

1. Where does it finish?
2. Which single translation replaces the three moves?

14 Are these triangles congruent? ★★★

Triangle \(ABC\) has \(A(0, 0)\), \(B(6, 0)\), \(C(0, 2)\); triangle \(DEF\) has \(D(1, 1)\), \(E(1, 7)\), \(F(-1, 1)\).

1. Show that \(\triangle ABC\cong\triangle DEF\) with \(A\to D\), \(B\to E\), \(C\to F\).
2. Describe a sequence of rigid motions that maps one onto the other.

15 Two parallel mirrors ★★★

Reflect \(P(-2, 3)\) in the line \(x = 1\), then reflect the image in the line \(x = 5\).

1. Find the final image.
2. Show that the composition is a translation and give its vector. Test it on \((0, -2)\).

16 Two mirrors that meet ★★★

1. Reflect \((x, y)\) in the x-axis, then in the y-axis. Which rotation is this?
2. Reflect in the line \(y = x\), then in the y-axis. Find the rule and the rotation. Test it on \((3, 4)\).

17 Finding the mirror line ★★★

A reflection maps \(A(1, 6)\) to \(A′(5, 2)\).

1. Find the equation of the line of reflection.
2. Is \(B(0, 1)\) moved by this reflection?

18 Glide reflection and orientation ★★★

Triangle \(RST\) has \(R(-4, 2)\), \(S(-1, 2)\), \(T(-3, 5)\). Reflect it in the x-axis, then translate by \(\langle 6, 1\rangle\).

1. Find the final vertices.
2. Compute the signed quantity \(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\) before and after. What does the sign change show?

19 Rotating about another center ★★★

Let \(C(2, 1)\) be the center. Find the image of (a) \(P(5, 4)\) by a \(90^\circ\) counterclockwise rotation; (b) \(Q(2, -3)\) by a \(180^\circ\) rotation; (c) \(R(6, 1)\) by a \(270^\circ\) counterclockwise rotation.

20 Why reflections keep distances ★★★

Let \(P(a, b)\) and \(Q(c, d)\) be any two points. Reflect them in the y-axis and prove that \(P′Q′ = PQ\). Then check with \(P(2, 3)\) and \(Q(-1, 7)\).

21 Symmetry detective ★★★

1. How many lines of symmetry does a regular octagon have, and what is its smallest rotation angle?
2. A logo has rotational symmetry of order 5. List all rotation angles (up to \(360^\circ\)) that map it onto itself.
3. A regular polygon has smallest rotation angle \(24^\circ\). How many sides does it have?

22 The Ferris wheel ★★★

A Ferris wheel has 12 equally spaced cars, numbered 1 to 12 going counterclockwise.

1. What is the smallest angle of rotation that maps the wheel onto itself?
2. Car 5 is moved by a \(210^\circ\) counterclockwise rotation. Which car position does it reach?
3. Where does car 10 go after a \(90^\circ\) counterclockwise rotation?