A gardener marks out a triangular raised bed, shape \(A\), with vertices \((1,1)\), \((3,1)\) and \((1,3)\), inside a square allotment plot fenced from \((0...
Question 1 Report
A gardener marks out a triangular raised bed, shape \(A\), with vertices \((1,1)\), \((3,1)\) and \((1,3)\), inside a square allotment plot fenced from \((0,0)\) to \((8,8)\), as shown. The gardener is choosing between two enlargements of shape \(A\), both centred at the origin \(O\).
- Find the coordinates of the image of shape \(A\) under an enlargement, centre \(O\), scale factor \(2\). (2)
- Determine whether enlarging shape \(A\), centre \(O\), scale factor \(3\) instead would fit entirely inside the fence. Give a reason. (2)
Answer Details
An enlargement centred at the origin multiplies every coordinate of every vertex by the scale factor; comparing the resulting vertices with the boundary of the fenced plot at \((8,8)\) then decides whether the enlarged shape stays inside the fence.
(a) Applying scale factor \(2\) to shape A's vertices \((1,1)\), \((3,1)\), \((1,3)\): \((2,2)\), \((6,2)\), \((2,6)\). [2 marks]
(b) Applying scale factor \(3\) instead gives \((3,3)\), \((9,3)\), \((3,9)\). Since the vertex \((9,3)\) has an x-coordinate of \(9\), which is greater than the fence's limit of \(8\), this enlarged image does not fit entirely inside the fenced plot. [2 marks]
Only one vertex needs to fall outside the boundary for the whole enlarged shape to fail to fit; here it is enough to spot that \(9\gt8\) for the vertex \((9,3)\), without needing to check every vertex in detail.
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