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Geometry and Measure - Position and movement (Transformations)

Grade 7IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Transformation: A process that changes the position, orientation, or size of a shape.

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Translation: Moving a shape without rotating or resizing it. Defined by a column vector.

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Reflection: Flipping a shape over a mirror line (axis of symmetry). Each point and its image are equidistant from the line.

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Rotation: Turning a shape around a fixed point called the center of rotation. Requires an angle and a direction (clockwise or anti-clockwise).

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Enlargement: Changing the size of a shape from a center of enlargement using a scale factor.

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Congruency: Shapes are congruent if they are the same size and shape (Translation, Reflection, Rotation).

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Similarity: Shapes are similar if they are the same shape but different sizes (Enlargement).

📐Formulae

Translation Vector: (xy)\begin{pmatrix} x \\ y \end{pmatrix} (where xx is horizontal movement and yy is vertical movement)

Scale Factor (kk): k=Image LengthObject Lengthk = \frac{\text{Image Length}}{\text{Object Length}}

Reflection in xx-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y)

Reflection in yy-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y)

Rotation 180∘180^\circ about origin: (x,y)→(−x,−y)(x, y) \rightarrow (-x, -y)

💡Examples

Problem 1:

A triangle with vertices A(1,2)A(1, 2), B(3,2)B(3, 2), and C(1,4)C(1, 4) is translated by the vector (2−3)\begin{pmatrix} 2 \\ -3 \end{pmatrix}. Find the new coordinates of A′A'.

Solution:

A′=(1+2,2−3)=(3,−1)A' = (1+2, 2-3) = (3, -1)

Explanation:

To translate a point, add the xx component of the vector to the xx-coordinate and the yy component of the vector to the yy-coordinate.

Problem 2:

Reflect the point P(4,5)P(4, 5) in the line y=xy = x.

Solution:

P′(5,4)P'(5, 4)

Explanation:

When reflecting in the line y=xy = x, the xx and yy coordinates are swapped.

Problem 3:

A square has a side length of 55 cm. It is enlarged by a scale factor of 33. What is the side length of the new square?

Solution:

5 cm×3=15 cm5 \text{ cm} \times 3 = 15 \text{ cm}

Explanation:

To find the new length after enlargement, multiply the original length by the scale factor kk.

Problem 4:

Rotate the point (2,0)(2, 0) 90∘90^\circ anti-clockwise about the origin (0,0)(0,0).

Solution:

(0,2)(0, 2)

Explanation:

A 90∘90^\circ anti-clockwise rotation moves a point from the positive xx-axis to the positive yy-axis.