Transformation, Congruency and Enlargement
REFERENCE: New General Mathematics for Senior Secondary Schools book 2, by M. F. Macrae and co.
OBJECTIVES: At the end of the lesson students should be able to;
>explain the meaning of transformation, congruency and enlargement.
>define and apply the properties of translation reflection and rotation.
>determine and use the scale factor of an enlargement.
>construct the enlargement of given figures.
>solve problems involving combine transformations.
CONTENT:
TRANSFORMATIONS
This is the change in presentation of the position or dimension of a given shape i.e representing the same figure in another way, either by changing its posture, ratio of the enlargement or position e.g (see fig15.1 on page 168) When a shape is transformed, the original shape is called the OBJECT while the redrawn/translated shape is called the IMAGE. A transformation is a mapping between two shape. In fig 15.1
where
A maps onto A’
B maps onto B’
C maps onto C’
hence triangle A’B’C’ is the image of triangle ABC.
CONGRUENCIES
Congruency is when corresponding dimension of two figures are identical i.e been alike in all ways e.g in the transformation of a shape, the OBJECT and the IMAGE are congruent shapes. There are three basic congruencies: translations, reflections and rotations.
Translations: this is a change of position in a straight line either vertical, horizontal or diagonal (see fig15.2 & fig15.3). The movement of the point can even be both diagonal and horizontal/ vertical or vertical and horizontal there by forming a pattern (see fig15.4)
When “translating” a shape or figure, every line or point movesmoves the same distance in the same direction by a fixed translation or displacement vector (see fig15.5 & the explanation below it).
See example 1 on page 16
The coordinate of point A as vertex of the quadrilateral is written as a single column vector. So that
A + (value of translation) ——> A’
B + (value of translation) ——> B’…..(see solution to the example)
as fig15.6 is showing the diagram of the OBJECT & the IMAGE(the translation of the original shape)
Reflections: On this let’s make use of the simple explanation as its given in the text book on page 170 – 171.
Rotation: This is the change in position by turning around see fig 15.17 on page172
The point where the shape or figure is turning about is called the “center of rotation“
When the rotation about the center of rotation is repeated on other centre of rotation, this will form a pattern.
When a point is rotated about its given centre of rotation in an anti clockwise direction, its angle is positive (see fig15.20) see also the explanation that follows fig15.20 on page172.
Let’s look into example 3 on page 173. The translation of P(2,4) to P’(-4,2) about the fix point O(0,0) is an anti clockwise movement (i.e positive) on 900 . Likewise example 4 on page 173, the movement of point T(-3,2) rotated through 1800 in an anticlockwise direction to translat its image at T’(3,-2)
REFLECTION PROPERTIES OF PLANE SHAPES
Plane shapes have line of symmetry or mirror line. The hand side is a reflection or mirrow image of the right-hand side.
When there are two lines of symmetry in a plane shape, it us said to be bilateral symmetry i.e one half matches the other.
The reflection properties of triangle will be given in summary thus:
1. The base angles of an isosceles triangle are equal.
2. The perpendicular from the vertex of an isosceles triangle to its base bisects the base and the vertical angle.
The reflection properties of circle will also be summarized thus:
1. The perpendicular from the centre of a circle to a chord bisects the chord.
OR
2. The perpendicular bisector of a chord passes through the centre of a circle.
(See page 174 for the explanations on the proofs)
ROTATION PROPERTIES
Rotation is a translation about a fixed point, which can be a given coordinate or the origin O. The rotation properties can be summarized thus:
Rotation properties of Parallelogram:
1. The opposite sides of a parallelogram are parallel.
2. The opposite sides of a parallelogram are equal.
3. The opposite angles of a parallelogram are equal.
4. A diagonal bisects a parallelogram.
5. The diagonals ofba parallelogram bisect one another. (See page 176 on explanation of the proof)
Rotation properties of Rhombus:
1. All four sides of a rhombus are equal.
2. That diagonals of a rhombus bisect one another at right angles.
3. The diagonals of a rhombus bisects its angles. (See page176-177 for the explanation on the proof)
ENLARGEMENT
Enlargement is a transformation in which a shape is magnified or diminished. Take a look at fig15.42 in the text book, the quadrilateral ABCD is translated to A1B1C1D1, A2B2C2D3 & A3B3C3D3. We can notice how the same shape is enlarged (increased in size and reduced as well) after the image A2B2C2D2, we noticed the translation diminished to a point(focus or centre of enlargement) at O and later increased in size to A3B3C3D3 (see page 178 the explanation and prove). The ratio of the original size/dimension to that of the image is said to be the “scale factor” which is the magnification of the object to the image. E.g see the diagram below.
Let’s look into example5 on page179, we where given the coordinates P(1,6),Q(5,4),R(3,2) for the original shape and P'(-2,-6),Q'(-4,-5),R'(-3,-4) for the image. After plotting the triangle, line |QR| intersects |Q’R’| at C as the centre of enlargement, hence the ratio of |CR’| to |CR| which is (-√8)÷(√32) = -1/2 or -0.5
Practice Questions
Page169, Exercise 15a, Nos 3-4.
Page171, Exercise 15b, Nos 7.
Page173, Exercise 15c, Nos 2&3.
Page177, Exercise 15e, Nos 3&5. Page179, Exercise 15f, Nos 1,2&3.