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Translating shapes

Learn how to draw the image of a given shape under a given translation.

Introduction

In this article, we'll practice the art of translating shapes. Mathematically speaking, we will learn how to draw the image of a given shape under a given translation.
A translation by open angle, a, comma, b, close angle is a transformation that moves all points a units in the x-direction and b units in the y-direction. Such a transformation is commonly represented as T, start subscript, left parenthesis, a, comma, b, right parenthesis, end subscript.

Part 1: Translating points

Let's study an example problem

Find the image A, prime of A, left parenthesis, 4, comma, minus, 7, right parenthesis under the transformation T, start subscript, left parenthesis, minus, 10, comma, 5, right parenthesis, end subscript.
A coordinate plane with point A at four, negative seven. The x- and y- axes scale by one.

Solution

The translation T, start subscript, left parenthesis, start color #01a995, minus, 10, end color #01a995, comma, start color #ca337c, 5, end color #ca337c, right parenthesis, end subscript moves all points start color #01a995, minus, 10, end color #01a995 in the x-direction and start color #ca337c, plus, 5, end color #ca337c in the y-direction. In other words, it moves everything 10 units to the left and 5 units up.
Now we can simply go 10 units to the left and 5 units up from A, left parenthesis, 4, comma, minus, 7, right parenthesis.
A coordinate plane with point A at four, negative seven. The x- and y- axes scale by one. A dashed arrow points ten units to the left and up five units to point A prime at negative six, negative two.
We can also find A, prime algebraically:
A, prime, equals, left parenthesis, 4, start color #01a995, minus, 10, end color #01a995, comma, minus, 7, start color #ca337c, plus, 5, end color #ca337c, right parenthesis, equals, left parenthesis, minus, 6, comma, minus, 2, right parenthesis

Your turn!

Problem 1

Draw the image of B, left parenthesis, 6, comma, 2, right parenthesis under transformation T, start subscript, left parenthesis, minus, 4, comma, minus, 8, right parenthesis, end subscript.

Problem 2

What is the image of left parenthesis, 23, comma, minus, 15, right parenthesis under the translation T, start subscript, left parenthesis, 12, comma, 32, right parenthesis, end subscript?
left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis

Part 2: Translating line segments

Let's study an example problem

Consider line segment start overline, C, D, end overline drawn below. Let's draw its image under the translation T, start subscript, left parenthesis, 9, comma, minus, 5, right parenthesis, end subscript.
A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one.

Solution

When we translate a line segment, we are actually translating all the individual points that make up that segment.
Luckily, we don't have to translate all the points, which are infinite! Instead, we can consider the endpoints of the segment.
A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one. An arrow points nine units to the right from C and down five units to point C prime. An arrow points nine units to the right from D and down five units to point D prime.
Since all points move in exactly the same direction, the image of start overline, C, D, end overline will simply be the line segment whose endpoints are C, prime and D, prime.
A coordinate plane with a line segment with endpoints C at negative seven, eight to D at negative four, one. The x- and y- axes scale by one. Another line segment has endpoints C prime at two, three and D prime at five, negative four. An arrow points from endpoint C to endpoint C prime and another arrow points from endpoint D to D prime.

Part 3: Translating polygons

Let's study an example problem

Consider quadrilateral E, F, G, H drawn below. Let's draw its image, E, prime, F, prime, G, prime, H, prime, under the translation T, start subscript, left parenthesis, minus, 6, comma, minus, 10, right parenthesis, end subscript.
A coordinate plane with a quadrilateral with vertices E at negative one, six, F at three, eight, G at two, two, and H at negative two, three. The x- and y- axes scale by one.

Solution

When we translate a polygon, we are actually translating all the individual line segments that make up that polygon!
A coordinate plane with a quadrilateral with vertices E at negative one, six, F at three, eight, G at two, two, and H at negative two, three. The x- and y- axes scale by one. A congruent quadrilateral with has vertices E prime at negative seven, negative four, F prime at negative three, negative two, G prime at negative four, negative eight, and H prime at negative eight, negative seven. An arrow points from vertex E to E prime. An arrow points from vertex F to F prime. Another arrow points from G to G prime, and an arrow points from vertex H to H prime.
Basically, what we did here is to find the images of E, F, G, and H and connect those image vertices.

Your turn!

Problem 1

Draw the image of triangle, I, J, K under the translation T, start subscript, left parenthesis, minus, 5, comma, 2, right parenthesis, end subscript.

Problem 2

Draw the images of start overline, L, M, end overline and start overline, N, O, end overline under the translation T, start subscript, left parenthesis, 10, comma, 0, right parenthesis, end subscript.

Challenge problem

The translation T, start subscript, left parenthesis, 4, comma, minus, 7, right parenthesis, end subscript mapped triangle, P, Q, R. The image, triangle, P, prime, Q, prime, R, prime, is drawn below.
Draw triangle, P, Q, R.

Want to join the conversation?

  • piceratops tree style avatar for user emanuelv04
    I dont understand that well without the graph
    (30 votes)
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  • piceratops seed style avatar for user fernsmith
    How do you construct a translation with a compass with a point away from the shape?
    (22 votes)
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  • blobby green style avatar for user benis
    This should be less challenging
    (13 votes)
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  • leafers seedling style avatar for user Felicia Nufor
    I need to understand problem 2 on translating line LM and NO
    (5 votes)
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    • winston baby style avatar for user limchi
      Here is a strategy:

      Move the green line onto the blue line so that you could see it overlapping it. Then move one point of the green line depending on the translation. In problem 2, the translation is (10,0), so it means that you have to move the point 10 units to the right, and 0 units up. Then do the same thing on the other point. Then you are done.
      (16 votes)
  • starky sapling style avatar for user jeffry
    why doesnt math slay
    (4 votes)
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  • primosaur seedling style avatar for user Destiny
    why was this made this is to hard curse new math
    (10 votes)
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    ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣦⣼⡏⠳⣜⢿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
    ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣷⣦⣤⣀⣀⣀⣴⣿⣿⣿⣿⣿⡿⠻⠆⠸⣎⣧⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
    ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⡄⠀⣿⢹⡇⢸⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
    (9 votes)
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  • starky sapling style avatar for user Justin772
    I was confused on the direction
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  • piceratops seedling style avatar for user DominicO
    this makes no sense
    (5 votes)
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  • scuttlebug green style avatar for user naniieatspeople
    How did I even get here? I thought this was supposed to teach me how to draw polygons with coordinates? GAHH MATH IS SO HARD!
    (4 votes)
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