Transformation

Transformation brings about a specific change in the position or size or both of a certain geometrical object on some definite rule.

Examples

Observe the figures and discuss the relationship of transformation. Try each one.

Mirror image (reflection)

The image is the same size and shape, but flipped. Both are at equal distance from the mirror line.

Clock hand (rotation)

The minute hand turns 6° each minute, clockwise, about the centre of the clock.

Balloon (enlargement)

The shape stays the same but the size changes by the same ratio in every direction.

Relationship of transformation

• In the mirror, the position changes (reflection).

• In the clock, the minute hand turns from 0 to 90° clockwise in 15 minutes (rotation).

• In the balloon, the size changes (enlargement).

Section Formula – Coordinates of a Point Dividing a Line Segment

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Isometric Transformation

A transformation in which only the position of the image changes compared to the original figure, but the shape and size remains the same, is called an isometric transformation. In this type, the object and its image are congruent. Reflection, rotation, and translation fall under this category.

Example

Triangle ABC (blue) and its image A′B′C′ (red). Choose a transformation.

Original figure Image

If P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) externally in the ratio m1:m2, then AP:BP = m1:m2

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Types of Transformation

Transformation is a process that moves an object to a new place, changes how big it is, or does both, following a definite rule. When a door is opened, only its position changes. When a balloon is inflated, its size changes.

Based on this, transformations are divided into two types:

  1. Isometric transformation
  2. Non-isometric transformation
Y Y' X X' O A(x₁, y₁) B(x₂, y₂) P(x, y) R S L M N

External Division

Suppose, in the figure, the line joining A(x1, y1) and B(x2, y2) is divided externally by P(x, y) in the ratio m1:m2.

Or, APBP = m1m2

Now, draw AL⊥OX, BM⊥OX, PN⊥OX, AR⊥PN and BS⊥PN

In the figure, AR = LN = ON − OL = x − x1,

BS = MN = ON − OM = x − x2

PR = PN − RN = PN − AL = y − y1

PS = PN − SN = PN − BM = y − y2

And APPB = m1m2

Right angled triangles ΔPAR and ΔBSP are similar.

ARBS = APBP [ ∵ The ratio of corresponding sides of similar triangle]

Or, x − x1x − x2 = m1m2

Or, m2(x − x1) = m1(x − x2)

Or, m2x − m2x1 = m1x − m1x2

Or, m1x2 − m2x1 = m1x − m2x

Or, m1x2 − m2x1 = x(m1 − m2)

Or, x = m1x2 − m2x1m1 − m2

Midpoint Formula

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If P(x, y) be the midpoint of the line segment joining the points A(x1, y1) and B(x2, y2), then AP = BP and AP:BP = m1:m2 = 1:1.

P(x, y) A(x₁, y₁) B(x₂, y₂)

Non-isometric Transformation

Non-isometric Transformation: A transformation in which the size of the image changes compared to the original figure is called non-isometric transformation. Enlargement falls under this category.

Example

Triangle ABC (blue) and its enlarged image A′B′C′ (red), centre of enlargement at the origin. Choose a scale factor.

Original figure Image

P(x, y) = (m1x2 + m2x1m1 + m2 , m1y2 + m2y1m1 + m2)

= (1.x2 + 1.x11 + 1 , 1.y2 + 1.y11 + 1)

= (x1 + x22 , y1 + y22)

∴ P(x, y) is called the coordinate of midpoint.