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Translation transformations

24.4 Translation transformations

Translating a point on the Cartesian plane

We can translate a point on the Cartesian plane from one position to another. For example, point \(A(1; 0)\) is translated 2 units to the right and 4 units upwards.

The position of point \(A\) after the transformation is called the image of \(A\) and we use the notation \(A'\) (we say "A prime").

image
the new position of a point, a line or an object after it has been transformed

\(A'\) is at \((3; 4)\).

\(A'\) is the image of \(A\) under a translation transformation.

Worked example 24.2: Translating a point on the Cartesian plane

Write down the coordinates of \(T'\) if \(T(2; 4)\) is translated \(3\) units to the left and \(2\) units downwards.

Complete the horizontal translation.

We move point \(T\) three units to the left (parallel to the \(x\)-axis). This translation moves \(T\) from the first quadrant to the second quadrant.

Complete the vertical translation.

We move point \(T\) two units down (parallel to the \(y\)-axis).

Write the final answer.

The coordinates of \(T'\) are \((-1; 2)\) and \(T'\) lies in the second quadrant.

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Translating a shape on the Cartesian plane

We can translate a shape on the Cartesian plane from one position to another. For example:

  • \(\triangle{ABC}\) is translated 2 units to the left and 2 units downwards
  • \(\triangle{A'B'C'}\) is the image of \(\triangle{ABC}\) under a translation transformation.

Notice that each vertex of \(\triangle{ABC}\) has been transformed by the same rule: slide 2 units to the left and 2 units downwards.

We can express this transformation rule as:

\[(x; y) \rightarrow (x - 2; y - 2)\]

This is called a mapping notation.

mapping notation
a representation of the rule that relates every point on the shape to the corresponding point on the image

Finding the image of a rectangle under a translation transformation

The diagram shows rectangle \(KLMN\) on the Cartesian plane.

  1. Draw a copy of rectangle \(KLMN\) on a Cartesian plane.

  2. Draw \(K'L'M'N'\), the image of \(KLMN\), translated 3 units to the right and 2 units upwards.

  3. Express this translation transformation in mapping notation.

  4. Compare the length of the corresponding sides of rectangle \(KLMN\) and rectangle \(K'L'M'N'\). What do you notice?

  5. Compare the size of the corresponding interior angles of rectangle \(KLMN\) and rectangle \(K'L'M'N'\). What do you notice?

  6. How would you describe these two shapes? Are they congruent shapes or similar shapes?

  7. Calculate the perimeter of rectangle \(KLMN\) and the perimeter of rectangle \(K'L'M'N'\).

  8. Calculate the area of rectangle \(KLMN\) and the area of rectangle \(K'L'M'N'\).

  9. A translation transformation produces congruent shapes which have the same area and the same perimeter. Is this statement true or false?

Worked example 24.3: Translating a triangle on the Cartesian plane

  1. Draw \(\triangle ABC\) with vertices \(A(-3; 2)\), \(B(3; 4)\) and \(C(-1; 6)\) on the Cartesian plane.
  2. Draw \(\triangle A'B'C'\), the image of \(\triangle ABC\), translated 4 units to the right and 3 units downwards.
  3. Write down the coordinates of the vertices \(A'\), \(B'\) and \(C'\).
  4. Express this translation transformation in mapping notation.

Draw the triangle on the Cartesian plane.

Translate each vertex of the triangle.

Translate each vertex of the \(\triangle ABC\) 4 units to the right and 3 units downwards to get the image \(\triangle A'B'C'\).

Write down the coordinates of the vertices of the image.

\[A'(1; -1), B'(7; 1) \text{ and } C'(3; 3)\]

Write the transformation in mapping notation.

\[(x; y) \rightarrow (x + 4; x - 3)\]
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