Translation, also known as slide, is a geometric transformation that moves every point of a figure the same distance and in the same direction. It is a fundamental concept in mathematics that is used to describe the movement of objects in space.
The concept of translation has been studied and used in mathematics for centuries. The ancient Greeks, such as Euclid and Pythagoras, were among the first to explore the properties of translations. However, it was not until the development of coordinate geometry in the 17th century that translations were formally defined and studied in a systematic way.
Translation is typically introduced in middle school or early high school mathematics curriculum. It is a concept that is covered in geometry courses and is an important topic for students to understand in order to solve geometric problems.
Translation involves several key knowledge points:
Vectors: A translation can be represented by a vector, which has both magnitude and direction. The vector represents the distance and direction that each point of the figure is moved.
Coordinate geometry: Translations can be described using coordinates. Each point of the figure is moved by adding or subtracting the same values to its x and y coordinates.
Congruence: Translations preserve the shape and size of the figure. The original figure and its translated image are congruent.
To perform a translation, follow these steps:
Determine the direction and distance of the translation. This can be done by analyzing the given information or by using vectors.
Apply the translation to each point of the figure. Add or subtract the same values to the x and y coordinates of each point, depending on the direction of the translation.
Plot the translated points to obtain the new figure.
There are two main types of translations:
Horizontal translation: In this type of translation, the figure is moved horizontally, either to the left or to the right.
Vertical translation: In this type of translation, the figure is moved vertically, either upwards or downwards.
Translations have several important properties:
Distance preservation: The distance between any two points on the original figure is the same as the distance between their corresponding points on the translated figure.
Angle preservation: The angles between any two lines on the original figure are the same as the angles between their corresponding lines on the translated figure.
Shape preservation: Translations preserve the shape of the figure. The original figure and its translated image are congruent.
To find or calculate a translation, you need to know the direction and distance of the movement. This can be determined by analyzing the given information or by using vectors.
If the direction and distance are given as vectors, simply add or subtract the vector values to the x and y coordinates of each point of the figure to obtain the translated points.
If the direction and distance are given as coordinates, add or subtract the same values to the x and y coordinates of each point of the figure to obtain the translated points.
The formula for translation can be expressed as:
(x', y') = (x + a, y + b)
where (x', y') are the coordinates of the translated point, (x, y) are the coordinates of the original point, and (a, b) are the direction and distance of the translation.
To apply the translation formula, simply substitute the values of (x, y) and (a, b) into the equation and calculate the coordinates of the translated point (x', y').
For example, if the original point has coordinates (2, 3) and the translation is 4 units to the right and 2 units upwards, the translated point can be calculated as:
(x', y') = (2 + 4, 3 + 2) = (6, 5)
The symbol or abbreviation for translation is "→" or "T".
There are several methods for performing a translation:
Using vectors: Represent the translation as a vector and add or subtract the vector values to the coordinates of each point.
Using coordinates: Add or subtract the same values to the x and y coordinates of each point to obtain the translated points.
Using graph paper: Draw the original figure on graph paper and slide it by counting the squares in the given direction and distance.
Example 1: Translate the point (3, 5) 2 units to the left and 3 units downwards.
Solution: Using the translation formula, we have:
(x', y') = (3 - 2, 5 - 3) = (1, 2)
Therefore, the translated point is (1, 2).
Example 2: Translate the triangle ABC with vertices A(1, 2), B(4, 3), and C(2, 5) 3 units to the right and 1 unit upwards.
Solution: Applying the translation formula to each vertex, we get:
A'(x', y') = (1 + 3, 2 + 1) = (4, 3) B'(x', y') = (4 + 3, 3 + 1) = (7, 4) C'(x', y') = (2 + 3, 5 + 1) = (5, 6)
Therefore, the translated triangle has vertices A'(4, 3), B'(7, 4), and C'(5, 6).
Example 3: Translate the line segment with endpoints P(2, 3) and Q(5, 7) 4 units to the left and 2 units downwards.
Solution: Applying the translation formula to each endpoint, we get:
P'(x', y') = (2 - 4, 3 - 2) = (-2, 1) Q'(x', y') = (5 - 4, 7 - 2) = (1, 5)
Therefore, the translated line segment has endpoints P'(-2, 1) and Q'(1, 5).
Translate the point (4, 6) 3 units to the right and 2 units upwards.
Translate the rectangle with vertices A(1, 2), B(4, 2), C(4, 5), and D(1, 5) 5 units to the left and 3 units downwards.
Translate the line segment with endpoints P(2, 3) and Q(5, 7) 2 units to the right and 4 units upwards.
Question: What is translation (slide)?
Answer: Translation, also known as slide, is a geometric transformation that moves every point of a figure the same distance and in the same direction. It is used to describe the movement of objects in space.