Need Help?

Get in touch with us

bannerAd

Translate Figures

Sep 10, 2022
link

Key Concepts

  • Define a translation, image and preimage.
  • Translate a figure in the coordinate plane.
  •  Understand isometry/ congruence transformation.
  •  Write a translation rule and verify congruence.
  • Understand translation theorem.

Introduction

Transformation moves or changes a figure in some way to produce a new figure called an image. Another name for the original figure is preimage.

Example: 1. In the given graph, A is the image. Find the preimage.

In the given graph, A is the image. Find the preimage

Translation

Translation moves every point of a figure to the same distance in the same direction. More specifically, a translation maps, or moves, the points P and Q of a plane figure to the points P΄ (read “P prime”) and Q΄, so that one of the following statements is true:

  • PP΄ = QQ΄ and  = , or
  • PP΄ = QQ΄ and  and  are collinear.
Translation

Translate a figure in the coordinate plane:

Translation of a figure in a coordinate plane happens horizontally and vertically.

W.r.t preimage On X-axis,

parallel

Positive translation – Shifting to the right (x + h)

Negative translation – Shifting to the left (x – h)

W.r.t preimage On Y-axis,

Positive translation – Shifting vertically upward (y + k)

Negative translation – Shifting vertically downward (y – k)

parallel

If (x, y) is the preimage, then, (x ± h, y ± k) is the image.

Example 1:

Graph quadrilateral ABCD with vertices A(–1, 2), B(-1, 5), C(4, 6), and D(4, 2). Find the image of each vertex after the translation (x, y) → (x + 3, y – 1). Then graph the image using prime notation.

Solution:

First, draw ABCD. Find the translation of each vertex by adding 3 to its x-coordinate and subtracting 1 from its y-coordinate. Then graph the image.

(x, y) → (x + 3, y – 1)

A(–1, 2) → A’(2, 1)

B(–1, 5) → B’(2, 4)

C(4, 6) → C’(7, 5)

D(4, 2) → D’(7, 1)

Example 1 solution

Congruence Transformation/ Isometry

An isometry is a transformation that preserves length and angle measure. Isometry is another word for congruence transformation.

Write a translation rule and verify congruence:

Let us understand this concept with the help of an example.

By using the graph, we go to image and pre-image of the figures and verify for congruence.

Example 2:

Write a rule for the translation of Δ ABC to Δ A’B’C’. Then verify that the transformation is an isometry.

Example 2

Solution:

To go from A to A’, move 4 units left and 1 unit up.

So, a rule for the translation is (x, y) → (x – 4, y + 1).

Use the SAS Congruence Postulate.

Notice that CB = C’B’ = 3, and AC = A’C’ = 2.

The slopes of  CB− = C’B’− are 0, and the slopes of CA− = C’A’−  are undefined,

So, the sides are perpendicular. Therefore, ∠ C and ∠ C’ are congruent right angles. So, Δ ABC ≅

Δ A’B’C’. The translation is an isometry.

Translation Theorem

To prove – Translation is an isometry.

Translation Theorem

Proof:

Given: P(a, b) and Q(c, d) are two points on a figure translated by (x, y) → (x + s, y + t).

The translation maps P(a, b) to P’(a + s, b + t) and Q(c, d) to Q’(c + s, d + t).

Use the Distance Formula to find PQ and P’Q’.

PQ = √(c−a)2+(d−b)2

√P′Q′ [c+s−(a+s)]2+[(d+t)−(b+t)2

= √(c+s−a−s)2+(d+t−b−t)2

  =√(c−a)2+(d−b)2

Use the Distance Formula to find PQ and P’Q’. 

PQ = √(c−a)2+(d−b)2

P′Q′= √P′Q′= [c+s−(a+s)]2+[(d+t)−(b+t)2

= √(c+s−a−s)2+(d+t−b−t)2

= √(c−a)2+(d−b)2

Therefore, PQ = P’Q’ by the Transitive Property of Equality.

Example 3:

Draw  with vertices A(2, 2), B(5, 2), and C(3, 5). Find the image of each vertex after the translation (x, y) → (x + 1, y + 2). Graph the image using prime notation.

Example 3

Solution:

Given vertices of a triangle are A(2, 2), B(5, 2), and C(3, 5).

Let us plot the vertices of a triangle on the graph.

Transformation rule:

(x, y) → (x + 1, y + 2)

A(2, 2) → A’(3, 4)

B(5, 2) → B’(6, 4)

C(3, 5) → C’(4, 7)

The image of each vertex after translation are:

A’(3,4)

B’(6,4)

C’(4,7)

Now, let us plot the image points of the triangle.

Example 3 solution

Example 4:

Δ ABC is the image of ABC after a translation. Write a rule for the translation.

Example 4

Solution:

On X-axis,

Positive translation – Shifting to the right (x + h)

Negative translation – Shifting to the left (x – h)

On Y-axis,

Positive translation – Shifting vertically upward (y + k)

Negative translation – Shifting vertically downward (y – k)

Here,

The figure moved to the right on the x-axis, and the figure moved upward on the y-axis.

Translation is 3 units right and 1 unit up.

Rule: (x, y) à (x + 3, y +1)

Exercise

  1. The image of (x, y) → (x + 4, y – 7) is  with endpoints P’(–3, 4) and Q’(2, 1). Find the coordinates of the endpoints of the preimage.
  2. ΔA’B’C’ is the image of ABC after a translation. Write a rule for the translation. Then verify that the translation is an isometry.
exercise 2
  • Use the translation (x, y) → (x – 8, y + 4). What is the image of B(21, 5)?
  • Use the translation (x, y) → (x – 8, y + 4). What is the preimage of C’(23, 10)?
  • The vertices of PQR are P(–2, 3), Q(1, 2), and R(3, –1). Graph the image of the triangle using prime notation. (x, y) → (x + 9, y – 2)
  • The vertices of PQR are P(–2, 3), Q(1, 2), and R(3, –1). Graph the image of the triangle using prime notation. (x, y) → (x – 2, y – 5)
  • Translate Q(0, –8) using (x, y) → (x – 3, y + 2).
  • The vertices of ABC are A(2, 2), B(4, 2), and C(3, 4). a. Graph the image of ABC after the transformation (x, y) → (x +1, y). Is the transformation an isometry?
  • The vertices of JKLM are J(–1, 6), K(2, 5), L(2, 2), and M(–1, 1). Graph its image after the transformation described. Translate 3 units left and 1 unit down.
  • The vertices of JKLM are J(–1, 6), K(2, 5), L(2, 2), and M(–1, 1). Graph its image after the transformation described. Translate 3 units right and 1 unit up.

Concept Map

What we have learned

  • Translating a figure in the coordinate plane.
  • Congruence transformation.
  •  Writing a translation rule and verifying congruence.
  •  Translation Theorem

Comments:

Related topics

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>
special right triangles_01

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […]

Read More >>
simplify algebraic expressions

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
solve right triangles

How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles.  Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>

Other topics