Translation is a geometric transformation that moves every point of a figure by the same distance in the same direction, without changing the shape's size, orientation, or internal angles. In mathematical terms, a translation is an isometry: a transformation that preserves distances and shape properties.
When you translate a shape, you are essentially "sliding" it to a new position on the coordinate plane. Every point on the original figure (called the pre-image) shifts by identical amounts in the horizontal and vertical directions to produce the new figure (called the image).
Mathematical Representation
For a translation in 2D space, if a point P has coordinates (x , y), and we translate it by a vector v=(tx ,ty), the new coordinates P′ will be:
ty = vertical displacement (positive = up, negative = down)
Types of Translation
Translation means sliding a figure from one position to another without rotating, flipping, or resizing it. The shape and orientation remain the same.
1) Horizontal Translation (Along X-axis)
A translation where only the x-coordinates change: v = (tx,0)
Example: Move point P(3, 4) by 5 units to the right.
P ′= (3+5,4+0) = (8,4)
2) Vertical Translation (Along Y-axis)
A translation where only the y-coordinates change: v = (0,ty)
Example: Move point Q(2, 3) by 4 units upward.
Q ′= (2+0,3+4) = (2,7)
3) Combined Translation (Both Horizontal and Vertical)
A translation with both x and y components: v=(tx,ty) where both tx ≠ 0 and ty ≠ 0.
Example: Move point R(1, 1) by 3 units right and 2 units down.
R ′ = (1+3,1−2) = (4,−1)
Properties of Translation
1. Congruence Preservation Translation produces a figure that is congruent to the original. This means:
Side lengths are preserved
Angles are preserved
Area and perimeter remain unchanged
2. Isometry Translation is an isometry transformation, which means:
Distance between any two points is preserved
The shape maintains its exact form
3. Vector Addition Property If you translate a shape by vector v1 = (a,b) and then by vector v2 = (c,d), it's equivalent to translating by the single vector v1+v2 = (a+c,b+d).
4. Inverse Translation For any translation by vector v, there exists an inverse translation by vector (−v) that returns the shape to its original position.
5. No Fixed Points Unlike rotation (which has a center) or reflection (which has an axis of symmetry), translation has no fixed points , every point in the plane moves.
Translating a Rectangle
Problem Statement
Consider a rectangle ABCD with vertices at: A(2,3) , B(6,3) , C(6,5) , D(2,5) . Translate this rectangle by the vector v = (3,2) (3 units to the right and 2 units upward).
Solution :
Apply the translation formula to each vertex:
For vertex A(2, 3):
A′ = (2+3,3+2) = (5,5)
For vertex B(6, 3):
B′ = (6+3,3+2) = (9,5)
For vertex C(6, 5):
C′ = (6+3,5+2) = (9,7)
For vertex D(2, 5):
D′ = (2+3,5+2) = (5,7)
The translated rectangle A'B'C'D' has vertices at (5, 5), (9, 5), (9, 7), and (5, 7), respectively. The shape maintains its original dimensions (width = 4 units, height = 2 units) and orientation.
Translation of Shapes - Solved Questions and Answers
Question 1 : Which of the following statements accurately describes the properties of translations of shapes?
(A) Translations preserve the angles of geometric figures (B) Translations change the area of the shape (C) Translations can rotate the shape (D) Translations increase the size of the shape
According to fundamental properties of translation
Translation is an isometric transformation, so all metric properties (distances, angles, area) are preserved. Only the position changes.
So, (A) is the correct option
Question 2 : Triangle ABC has vertices A(1, 2), B(4, 2), and C(2, 5). Translate it by vector v = (−2,3). What are the coordinates of the image triangle A'B'C'?
Solution :
For each vertex, apply: (x′,y′) = (x+tx , y+ty) where v = (−2,3)
For A(1, 2): A′ = (1+(−2),2+3) = (−1,5)
For B(4, 2): B′ = (4+(−2),2+3) = (2,5)
For C(2, 5): C′ = (2+(−2),5+3) = (0,8)
Verification:
Distance AB = √(4−1)2 + (2−2)2 = 3 units
Distance A'B' = √(2−(−1))2+(5−5)2 = 3 units
Distances are preserved, confirming this is a valid translation.
Question 3 : Shape X is shown on a coordinate grid. Shape Y appears to be a translation of Shape X. Which column vector BEST describes the translation from X to Y?
Solution :
A point on X is at (3, 4) and the corresponding point on Y is at (7, 1).
Question 4 : A shape is translated by 4 units to the right and 3 units upward. Which of the following is NOT a property of the translated shape compared to the original ?
(A) Congruent to the original shape (B) Same perimeter as the original (C) Same orientation as the original (D) Same distance from the origin
Solution :
Since the shape moves, its distance from the origin changes. This is the ONLY property among the options that is NOT preserved by translation. So , (D) is the correct option.
Question 1 : A parallelogram PQRS has vertices at P(1, 1), Q(5, 1), R(6, 3), and S(2, 3). Translate it by the vector v=(2,4).
Find: (a) The coordinates of the translated parallelogram P'Q'R'S' (b) Calculate the distance between P and P'
Question 2 : Two triangles, Triangle 1 and Triangle 2, appear to have the same shape. Triangle 1 has vertices at (0, 0), (4, 0), and (2, 3). Triangle 2 has vertices at (5, 2), (9, 2), and (7, 5).
Determine: (a) Is Triangle 2 a translation of Triangle 1? Show your work. (b) If yes, what is the translation vector? (c) If no, what transformation would map Triangle 1 to Triangle 2?
Question 3 : Point P(2, 3) is translated by vector (4, -1). Find the coordinates of P'.
Question 4 : Point Q(-2, 5) moves to Q'(1, 2) after translation. Find the translation vector and apply it to point R(0,0).
Question 5 : Square with corners (3,3), (6,3), (6,6), (3,6) is translated 1 unit left and 2 units down. Find all new corner coordinates.
