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High school geometry

Course: High school geometry > Unit 1

Lesson 3: Translations
Math
High school geometry
Performing transformations
Translations
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Determining translations

Google Classroom
Learn how to find the necessary translation to map a given source shape onto a given image shape.
In this article, we will solve problems where we are given starting and ending coordinates and asked to figure out what translation must have occurred.

Part 1: Determining the translation for a single pair of points

Let's study an example problem

A translation maps point A(3,7) to point A′(6,−2). Let's determine what translation this is.

Solution

Step 1: Horizontal shift. A is shifted 3 units to the right because (6)−(3)=+3.
Step 2: Vertical shift. A is shifted 9 units down because (−2)−(7)=−9.
The answer: A is mapped onto A′ under a translation by ⟨3,−9⟩.

Your turn!

Problem 1

Determine the translation that maps point B(2,1) to point B′(−4,5).
⟨
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
⟩

Problem 2

Determine the translation that maps point C(7,5) to point C′(5,5).
⟨
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
⟩

Problem 3

In general, which calculation gives the exact vertical shift of a translation from point P to point P′?
Choose 1 answer:

Challenge problem

A certain translation takes point D(−3,10) to point D′(−12,21).
What is the image of E(17,−9) under this translation?
(
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
)

Part 2: Determining the translation for a pair of polygons

Let's study an example problem

Consider the quadrilaterals drawn below. Let's determine the translation that maps the pre-image FGHI onto the image F′G′H′I′.
A coordinate plane. The x- and y-axes both scale by one. Trapezoid FGHI with point F at negative four, six, point G at one, six, point H at zero, two, and point I at negative four, two. Trapezoid F prime, G prime, H prime, I prime with points F prime at two, three, G prime at seven, three, H prime at six, negative one, and I prime two, negative one.

Solution

Let's focus in on a pair of corresponding points, such as F(−4,6) and F′(2,3). If we can find the translation that takes F to F′, we will necessarily know the translation that takes the entire pre-image quadrilateral to its image!
Horizontal shift: (2)−(−4)=+6
Vertical shift: (3)−(6)=−3
Therefore, FGHI is mapped onto F′G′H′I′ under a translation by ⟨6,−3⟩.

Your turn!

Determine the translation that maps △JKL onto △J′K′L′.
A coordinate plane with a triangle with vertices J at two, negative four, K at eight, negative three, and L at six, negative eight. The x- and y- axes scale by one. A congruent triangle has vertices J prime at negative two, three, K prime at four, four, and L prime at two, negative one.
⟨
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
⟩

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