Example 1 Translate a Figure

Example 2 Find a Translation Matrix

Example 3 Dilation

Example 4 Reflection

Example 5 Rotation

Find the coordinates of the vertices of the image of quadrilateral ABCD with A(–5, –1), B(–2, –1), C(–1, –4), and D(–3, –5) if it is moved 3 units to the right and 4 units up. Then graph ABCD and its image ABCD.

Write the vertex matrix for quadrilateral ABCD.

To translate the quadrilateral 3 units to the right, add 3 to each x-coordinate.

To translate the figure 4 units up, add 4 to each y-coordinate.

This can be done by adding the translation matrix

to the vertex matrix of ABCD.

Vertex Matrix Translation Vertex Matrix of ABCD Matrix of A'B'C'D'

The coordinates of ABCD are A(–2, 3), B(1, 3), C(2, 0), D(0, –1).

Graph the preimage and the image. The two graphs have the same size and shape.

Answer: A(–2, 3), B(1, 3), C(2, 0), D(0, –1)

Find the coordinates of the vertices of the image ofquadrilateral HIJK with H(2, 3), I(3, –1), J(–1, –3), and K(–2, 5) if it is moved 2 units to the left and 2 units up. Then graph HIJK and its image HIJK.Answer: H(0, 5), I(1, 1),

J(–3, –1), K(–4, 7)

Short-Response Test Item Rectangle EFGH is the result of a translation of the rectangle EFGH. A table of the vertices of each rectangle is shown. Find the coordinates of F and G.

Rectangle EFGH Rectangle EFGHE(–2, 2) E(–5, 0)

F F(1, 0)

G(4, –2) GH(–2, –2) H(–5, –4)

Read the Test Item

You are given the coordinates of the preimage and image of points E and H. Use this information to find the translation matrix. Then you can use the translation matrix to find the coordinates of F and G.

Solve the Test Item

Write a matrix equation. Let (a, b) represent the coordinates of F and let (c, d) represent the coordinates of G.

Since these two matrices are equal, corresponding elements are equal.

Solve an equation for x. Solve an equation for y.

Use the values for x and y to find the variables forF(a, b) and G(c, d).

Answer: So, the coordinates of F are (4, 2) and the coordinates of G are (1, –4).

Short-Response Test Item Rectangle ABCD is the result of a translation of the rectangle ABCD. A table of the vertices of each rectangle is shown. Find the coordinates of D and A.

Rectangle ABCD Rectangle ABCDA(–4, 5) AB(–1, 5) B(8, 0)

C(–1, 0) C(8, –5)

D D(5, –5)

Answer: The coordinates of D are (–4, 0) and the coordinates of A are (5, 0).

XYZ has vertices X(1, 2), Y(3, –1), and Z(–1, –2). Dilate XYZ so that its perimeter is twice the original perimeter. What are the coordinates of the vertices of XYZ?

If the perimeter of a figure is twice the original figure, then the lengths of the sides of the figure will be twice the measure of the original lengths. Multiply the vertex matrix by the scale factor of 2.

The coordinates of the vertices of XYZ are X(2, 4), Y(6, –2), and Z(–2, –4).

Graph XYZ and XYZ. The triangles are not congruent. The image has sides that are half the length of those of the original figure.

Answer: X(2, 4), Y(6, –2), Z(–2, –4)

ABC has vertices A(2, 1), B(–3, –2), and C(1, 4). Dilate ABC so that its perimeter is four times the original perimeter. What are the coordinates of the vertices of ABC?

Answer: A(8, 4), B(–12, –8), C(4, 16)

Find the coordinates of the vertices of the image of pentagon PENTA with P(–3, 1), E(0, –1), N(–1, –3), T(–3, –4), and A(–4, –1) after a reflection across the x-axis.

Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the x-axis.

Answer: The coordinates of the vertices of PENTA areP(–3, –1), E(0, 1), N(–1, 3),T(–3, 4), and A(–4, 1). Noticethat the preimage and image arecongruent. Both figures have thesame size and shape.

Find the coordinates of the vertices of the image of pentagon PENTA with P(–5, 0), E(–3, 3), N(1, 2), T(1, –1), and A(–4, –2) after a reflection across the line y = x.

Answer: P(0, –5), E(3, –3), N(2, 1), T(–1, 1), A(–2, –4)

Find the coordinates of the vertices of the image of DEF with D(4, 3), E(1, 1), and F(2, 5) after it is rotated 90 counterclockwise about the origin.

Write the ordered pairs in a vertex matrix. Then multiply the vertex matrix by the rotation matrix.

Answer: The coordinates of the vertices of triangle DEF are D(–3, 4), E(–1, 1), and F (–5, 2). The image is congruent to the preimage.

Find the coordinates of the vertices of the image of TRI with T(–1, 2), R(–3, 0), and I(–2, –2) after it is rotated 180 counterclockwise about the origin.

Answer: T (1, –2), R(3, 0), I(2, 2)