الرياضيات (بالإنجليزية) الصف الثالث الإعدادي، الأسبوع 9: Dilation in the coordinate plane about the origin: (x, y) → (kx, ky)

Dilation

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

Dilation in the coordinate plane about the origin: (x, y) → (kx, ky)

Mathematics Prep 3 Term 1, Unit 3, Lesson 3

GoalFind the image of a point or a figure under a dilation with centre the origin, undo a dilation, and see how areas change.

New words

origin · the point (0, 0), where the axes meetcoordinates · the pair (x, y) that gives a point’s placerule (x, y) → (kx, ky) · every coordinate is multiplied by kundo · apply the factor 1 ÷ k to get back to the original

Explanation

When the centre is the origin (0, 0), a dilation with factor k has a very simple rule: (x, y) → (kx, ky). Each coordinate is multiplied by k. Check: the point (kx, ky) lies on the ray from the origin through (x, y), and its distance from the origin is k times as long. To find the image of a polygon, change each vertex and join the images. To undo a dilation with factor k apply the factor 1 ÷ k. Because every length is multiplied by k, the area of a figure is multiplied by k × k = k². For example, a rectangle of area 6 dilated with k = 4 has area 16 × 6 = 96.

Worked examples

Find the image of △PQR with P(−2, 4), Q(6, 2), R(0, −3) under a dilation with the origin as centre and k = 1/2. Multiply each coordinate by 1/2: P′(−1, 2), Q′(3, 1), R′(0, −1.5). The picture shows the original triangle and its half-size image, with the same shape and parallel sides.

A point P′(6, 9) is the image of P under a dilation about the origin with k = 3. Find P. Undo it with 1 ÷ 3: P = (6 ÷ 3, 9 ÷ 3) = (2, 3). Check: 3 × (2, 3) = (6, 9) ✓.

The triangle with vertices (1, 1), (3, 1), (2, 3) is dilated about the origin with k = 3. Find the image and compare the areas. The vertices become (3, 3), (9, 3), (6, 9). Area of the original = 1/2 × 2 × 2 = 2. Area of the image = 1/2 × 6 × 6 = 18 = 9 × 2 = 3² × 2.

Class activity

Grid enlargement. On squared paper plot a small house shape (a square with a triangle roof) near the origin. Multiply all coordinates by 2 and draw the new house. Then multiply the first house’s coordinates by 1/2. Compare the three drawings.

Practice

Find the image of (4, −6) under a dilation about the origin with k = 1/2.

Find the image of (−2, 5) with k = 3.

A point has image (6, 9) with k = 3. Which point was dilated?

Which factor gives a reduction: 120%, 50%, 100%, 250%?

The triangle (0, 0), (4, 0), (0, 3) is dilated with k = 2. Find the image and both areas.