A site engineer marks the corners of a shed's foundation as triangle \(P\) on a grid, with each unit representing one metre. Overnight, a digger accidentall...
Question 1 Report
A site engineer marks the corners of a shed's foundation as triangle \(P\) on a grid, with each unit representing one metre. Overnight, a digger accidentally shifts every marker peg by the vector \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\), leaving triangle \(Q\) in the position shown below.
- Find the coordinates of the vertices of triangle \(P\) before the digger moved the pegs. (3)
- Write down the column vector of the translation that would move triangle \(Q\) back onto triangle \(P\). (1)
Answer Details
A translation slides every point of a shape by the same vector, so if the vector \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) maps triangle \(P\) onto triangle \(Q\), then every vertex of \(Q\) is found by adding \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) to the corresponding vertex of \(P\). To work backwards from \(Q\) to \(P\), the same vector is subtracted from each vertex of \(Q\).
(a) Reading the vertices of \(Q\) from the grid gives \((3, 4)\), \((6, 4)\) and \((3, 6)\). Subtracting \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) from each:
- \((3, 4) - (2, 3) = (1, 1)\)
- \((6, 4) - (2, 3) = (4, 1)\)
- \((3, 6) - (2, 3) = (1, 3)\)
So triangle \(P\) had vertices \((1, 1)\), \((4, 1)\) and \((1, 3)\). One mark is available for each correctly stated vertex. [3]
(b) A translation is undone by the vector of the same magnitude in the opposite direction, so the column vector that maps \(Q\) back onto \(P\) is \(\begin{pmatrix} -2 \\ -3 \end{pmatrix}\). [1]
A common error is to add the vector to \(Q\) instead of subtracting it when reversing a translation; always check which shape is the object and which is the image before deciding whether to add or subtract the vector.
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