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

1 Applying the rules / 4 pts

Let \(P(-6, 2)\). Find the image of \(P\) under: (a) the translation by \(\langle 7, -5\rangle\); (b) the reflection in the y-axis; (c) the reflection in the x-axis; (d) the rotation of \(90^\circ\) counterclockwise about the origin.

2 A half turn / 3 pts

Triangle \(KLM\) has \(K(1, 1)\), \(L(4, 1)\), \(M(1, 3)\).

1. Rotate it \(180^\circ\) about the origin.
2. Show that \(LM = L′M′\).

3 Composition / 4 pts

Let \(Q(3, -2)\), let \(T\) be the translation by \(\langle -1, 4\rangle\) and \(R\) the reflection in the x-axis.

1. Find the image of \(Q\) by \(T\) then \(R\).
2. Find the image of \(Q\) by \(R\) then \(T\).
3. Are the two results the same? What does it show?

4 Identify the rotation / 3 pts

Triangle \(ABC\) with \(A(1, 2)\), \(B(4, 2)\), \(C(1, 5)\) is mapped to \(A′(-2, 1)\), \(B′(-2, 4)\), \(C′(-5, 1)\) by a rotation about the origin.

1. Find the rule of the rotation.
2. Verify it for \(B\) and \(C\).
3. Which clockwise angle gives the same image?

5 Symmetry / 3 pts

1. A regular nonagon has 9 sides. How many lines of symmetry does it have, and what is its smallest rotation angle?
2. A rhombus that is not a square: give its number of lines of symmetry and its smallest rotation angle.

6 Congruent triangles / 3 pts

Triangle \(ABC\): \(A(0, 0)\), \(B(3, 0)\), \(C(0, 4)\). Triangle \(GHI\): \(G(2, 2)\), \(H(2, 5)\), \(I(-2, 2)\).

1. Show that the triangles are congruent.
2. Describe two rigid motions that map \(ABC\) onto \(GHI\).