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Congruency, Enlargement and Combined Transformations · Form 5

Congruency, Enlargement and Combined Transformations: Worked Examples (Medium)

Two- and three-step problems: turning an area ratio into a scale factor, enlarging from a given centre, and carrying out a combined transformation in the correct order. For students already comfortable with single transformations.

Written by the spmmath.com.my editorial team.

Worked example 1

An enlargement maps a triangle with area 5 cm² onto an image with area 45 cm². (a) Find the scale factor k (k > 0).

(b) One side of the object is 4 cm; find the length of the corresponding side of the image.

  1. Area ratio = area of image ÷ area of object = 45 ÷ 5 = 9.
  2. The area ratio equals k², so k² = 9.
  3. Take the positive square root: k = √9 = 3.
  4. Corresponding side = k × object side = 3 × 4 = 12 cm.

Worked example 2

The point P(5, 2) undergoes an enlargement with centre C(1, 0) and scale factor 2. Find the coordinates of the image P′.

  1. Image = centre + k × (point − centre); measure the distance from the centre C, not from the origin.
  2. Vector from C to P: (5 − 1, 2 − 0) = (4, 2).
  3. Multiply by k = 2: (2 × 4, 2 × 2) = (8, 4).
  4. Add back the centre: (1 + 8, 0 + 4) = (9, 4).

Worked example 3

The point A(4, 2) is first rotated 90° clockwise about the origin to give A′, and then A′ is reflected in the y-axis to give A″. Find the coordinates of A″.

  1. A 90° clockwise rotation about the origin sends (x, y) → (y, −x).
  2. So A′ = (2, −4).
  3. Reflection in the y-axis sends (x, y) → (−x, y).
  4. So A″ = (−2, −4).

Worked example 4

P(1, 4) undergoes an enlargement with centre C(−1, 2) and scale factor 3. Find the coordinates of the image.

  1. Image = C + k(P − C)
  2. P − C = (1 − (−1), 4 − 2) = (2, 2)
  3. k(P − C) = 3 × (2, 2) = (6, 6)
  4. Image = (−1 + 6, 2 + 6) = (5, 8)

Worked example 5

Point A(3, 1) is first translated by the vector (−1, 2), then rotated 90° anticlockwise about the origin O. Find the coordinates of the final image A″.

  1. Translation by (−1, 2): A(3, 1) → (3 − 1, 1 + 2) = (2, 3)
  2. Rotation 90° anticlockwise about O: (x, y) → (−y, x)
  3. (2, 3) → (−3, 2)

Worked example 6

P(3, −1) undergoes an enlargement with centre O and scale factor −2. Find the coordinates of the image.

  1. Enlargement centre O, factor k = −2: (x, y) → (kx, ky)
  2. P(3, −1) → (−2 × 3, −2 × (−1)) = (−6, 2)

Worked example 7

Point E(2, −5) is first reflected in the x-axis to give E′, and then E′ is translated by the vector (3, 4). Find the coordinates of the final image E″.

  1. Reflection in the x-axis: E(2, −5) → E′(2, 5), the y-coordinate changes sign
  2. Translation by (3, 4): E′(2, 5) → E″(2 + 3, 5 + 4) = E″(5, 9)

Worked example 8

An enlargement maps a quadrilateral of area 12 cm² onto an image of area 108 cm². (a) Find the scale factor of the enlargement, k (k > 0).

(b) The corresponding side of the image is 15 cm long; find the length of the object's side.

  1. Ratio of areas = image area ÷ object area = 108 ÷ 12 = 9
  2. k² = 9, so k = 3
  3. Length of object's side = length of image's side ÷ k = 15 ÷ 3 = 5 cm

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What do medium-level questions add to this transformations topic?

Medium questions ask you to perform two transformations one after another (such as a translation followed by an enlargement) and find the final image, or to describe a single transformation that has the same effect as two combined ones. You also need to relate the linear scale factor to the area scale factor (area scale factor = k²) when enlargement is involved.

What order mistake happens with combined transformations?

Applying the two transformations in the wrong order, doing the second transformation first, usually gives a different (wrong) final image, since transformations are generally not interchangeable in sequence. Always follow the order stated in the question exactly, and find the image of the first transformation before applying the second to that image.

What error happens when using the area scale factor?

Students sometimes use the linear scale factor directly to find a new area instead of squaring it first, or square an area that was already given (rather than a length), giving a badly wrong answer. Remember area scale factor equals the square of the linear scale factor, and check which quantity, length or area, is given before applying it.

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