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Dilation

الأسبوع 9 · التمدد: مركز التمدد ومعامل التمدد والتكبير والتصغير وصورة نقطة أو مضلع في المستوى الإحداثي · 3 دروس
1

Dilation with a centre and a positive factor k: enlargement and reduction

Mathematics Prep 3 Term 1, Unit 3, Lesson 3
GoalFind the image of a point under a dilation, tell an enlargement from a reduction, and find k from two lengths.
New words
dilation · a transformation that enlarges or reduces a figure from a fixed centrecentre of dilation · the fixed point C: it is its own imagedilation factor k · image distance from the centre ÷ original distance from the centreimage · the point or figure after the dilation; P′ is the image of P
Explanation

A dilation needs a centre C and a positive number k. The image of a point P is the point P′ on the ray CP with CP′ = k × CP. The centre stays where it is. If k > 1 the figure is enlarged; if 0 < k < 1 it is reduced; if k = 1 nothing changes. So k = CP′ ÷ CP, and also k = A′B′ ÷ AB for any segment and its image. The image is similar to the original: the angles are unchanged, every image side is k times the original side, and each image side is parallel to its original side. Measure distances from the centre, not between P and P′. A factor given as a percentage is written as a number: 120% is 1.2 (enlargement) and 50% is 0.5 (reduction).

OABCA′B′C′
Worked examples
In the picture O is the centre and k = 2. A(2, 1) goes to A′(4, 2), B(4, 1) to B′(8, 2), C(3, 3) to C′(6, 6). Check that the images lie on the rays: A′ is twice as far from O as A (the point (4, 2) is twice (2, 1)). Compare sides: AB = 2 and A′B′ = 4 = 2 × 2. AB and A′B′ are both horizontal, so they are parallel.
Find the images: (a) OP = 6 cm, k = 3; (b) OQ = 10 cm, k = 2/5. (a) OP′ = 3 × 6 = 18 cm: an enlargement. (b) OQ′ = 2/5 × 10 = 4 cm: a reduction, since 0 < 2/5 < 1.
A point is 6 cm from the centre and its image is 9 cm from the centre. Find k and the factor that undoes the dilation. k = 9 ÷ 6 = 3/2 (enlargement). To go back from the image to the original we need 6 ÷ 9 = 2/3, which is 1 ÷ k.
Class activity

Shadow on the wall. In a dark room, shine a torch (the centre) on a cut-out paper shape so that its shadow falls on a wall. Move the torch and measure: the distance from the torch to the shape and to the wall, and the sizes of the shape and its shadow. Is the shadow a dilation of the shape? What is k?

Practice
1
k = 4 and OP = 3 cm. Find OP′.
2
k = 1/3 and OP = 12 cm. Find OP′. Is it an enlargement or a reduction?
3
OP = 5 cm and OP′ = 20 cm. Find k.
4
A segment AB = 6 cm is dilated with k = 0.5. Find A′B′ and say how it lies compared with AB.
5
Which factors give an enlargement: 0.8, 1, 1.25, 150%, 40%?