IMO Practice Test — Transformations
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
Point \(A(1, 2)\) is reflected across the line \(y = x\), and then the intermediate image is translated by vector \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\). Find the final coordinates.
\((5, 0)\)
\((4, 2)\)
\((5, 1)\)
\((4, 0)\)
Explanation: Reflection over \(y=x\) swaps values: \((1, 2) \rightarrow (2, 1)\). Vector shift: \((2+3, 1-1) = (5, 0)\).
Question 2 of 6
A geometric shape has a rotational symmetry of order \(4\). If the area of this shape is \(24\text{ cm}^2\) and it is enlarged by a scale factor of \(k = 0.5\), find the area of the transformed image.
\(6\text{ cm}^2\)
\(12\text{ cm}^2\)
\(48\text{ cm}^2\)
\(96\text{ cm}^2\)
Explanation: New area \(= \text{Old area} \times k^2 = 24 \times (0.5)^2 = 24 \times 0.25 = 6\text{ cm}^2\).
Question 3 of 6
Point \(Q(2, 3)\) is rotated \(90^\circ\) counter-clockwise around the origin, then reflected across the \(y\)-axis. What are the final coordinates of the image?
\((3, 2)\)
\((-3, 2)\)
\((3, -2)\)
\((-2, 3)\)
Explanation: Counter-clockwise \(90^\circ\): \((x,y)\rightarrow(-y,x) \implies (-3,2)\). Reflect over \(y\)-axis flips \(x\)-sign: \(\rightarrow (3,2)\).
Question 4 of 6
A regular polygon has an angle of rotational symmetry measuring exactly \(40^\circ\). How many lines of symmetry does this polygon contain?
6
8
9
12
Explanation: \(\text{Order} = 360 \div 40 = 9\). For regular polygons, lines of symmetry equal the order, which is \(9\).
Question 5 of 6
A triangle with vertices at \((0,0)\), \((4,0)\), and \((0,6)\) is enlarged from a center at the origin. If the area of the image triangle is \(108\text{ units}^2\), find the scale factor \(k\).
\(k = 3\)
\(k = 4\)
\(k = 9\)
\(k = 27\)
Explanation: Original area \(= \frac{1}{2} \times 4 \times 6 = 12\). Area ratio \(= 108 \div 12 = 9\). Since \(k^2 = 9\), the scale factor is \(k = 3\).
Question 6 of 6
A shape is translated by vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\), then reflected over the line \(y = 0\) (\(x\)-axis), ending at point \((5, -1)\). What were the coordinates of the original starting point?
\((3, 4)\)
\((3, -2)\)
\((7, 2)\)
\((3, 2)\)
Explanation: Undo reflection over \(x\)-axis first: \((5, 1)\). Then subtract the translation vector: \((5-2, 1-(-3)) = (3, 4)\).