Teach: Plane Figures ③ Translations

Plane Figures ③ Translations

Moving a figure without changing its shape or size is called a transformation. There are three types: translation, rotation, and reflection. First, let us study the most basic type, translation.

What Is a Translation?

Moving a figure by sliding it a fixed distance in a fixed direction is called a translation.
　・The figure’s shape and size do not change (the figures are congruent).
　・The segments connecting corresponding points are all parallel and equal in length.

Properties of Translations

When △ABC is translated to become △A'B'C':
　① AA' // BB' // CC' (the segments connecting corresponding points are all parallel)
　② AA' ＝ BB' ＝ CC' (their lengths are all equal)
　③ AB ＝ A'B', BC ＝ B'C', CA ＝ C'A' (the lengths of corresponding sides do not change)
　④ ∠A ＝ ∠A', ∠B ＝ ∠B', ∠C ＝ ∠C' (the measures of corresponding angles do not change)

Example ①
△ABC was translated 5 cm in the direction shown by the arrow to create △A'B'C'. Answer the following questions.
(1) What is the length of AA'?
(2) What is the length of BB'?
(3) Is AC // A'C' true?
Solution:
(1) 5 cm　(2) 5 cm (all segments connecting corresponding points have the same length)
(3) In a translation, the directions of the sides do not change, so AC and A'C' are parallel → Yes

Coordinates and Translations

The point obtained by translating (x, y) by a units in the x-axis direction and b units in the y-axis direction is (x + a, y + b).
　Example: Translating point A(2, 3) by 4 in the x-axis direction and −1 in the y-axis direction gives A'(6, 2).

Example ②
Find the coordinates of point P' obtained by translating point P(−1, 4) by 3 in the x-axis direction and −2 in the y-axis direction.
Solution:
P' = (−1 + 3, 4 + (−2)) = (2, 2)
Key Points
  • Translation: sliding a figure. Its shape and size remain unchanged.
  • The segments connecting corresponding points are all parallel and have the same length.
  • Coordinates: (x, y) → (x+a, y+b).
  • Corresponding sides are parallel, and corresponding angles are equal.

Practice Problems

  1. When △ABC is translated in the direction AA' = 4 cm to create △A'B'C', find the length of CC'.
  2. Find the coordinates of point A' obtained by translating A(3, −2) by −5 in the x-axis direction and 4 in the y-axis direction.
  3. Select all the correct statements about translations.
    　ア) The orientation of the figure changes.　イ) The size of the figure changes.　ウ) Corresponding sides become parallel.　エ) Corresponding angles are equal.

Answers and Explanations

  1. Answer: 4 cm
    Explanation: In a translation, all segments connecting corresponding points have equal lengths.
  2. Answer: (−2, 2)
    Explanation: (3 + (−5), −2 + 4) = (−2, 2).
  3. Answer: ウ、エ
    Explanation: A translation changes neither orientation nor size. Corresponding sides remain parallel and equal in length, and angles remain unchanged.

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