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The diagram shows a triangle P on a grid - Edexcel - GCSE Maths - Question 12 - 2020 - Paper 3
Question 12
The diagram shows a triangle P on a grid. Triangle P is rotated 180º about (0, 0) to give triangle Q. Triangle Q is translated by (-5, -2) to give triangle R. (a) ... show full transcript
Worked Solution & Example Answer:The diagram shows a triangle P on a grid - Edexcel - GCSE Maths - Question 12 - 2020 - Paper 3
Step 1
Describe fully the single transformation that maps triangle P onto triangle R.
Answer
To determine the single transformation mapping triangle P onto triangle R, we start by analyzing the transformations that occur.
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Rotation: Triangle P is rotated 180º about the origin (0, 0) to produce triangle Q. This transformation inverts the coordinates of each vertex of triangle P:
- If a vertex of triangle P is at (x, y), after rotation, it becomes (-x, -y).
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Translation: Triangle Q is further translated by the vector (-5, -2) to produce triangle R. This means that we subtract 5 from the x-coordinates and 2 from the y-coordinates of each vertex of triangle Q:
- If a vertex of triangle Q is at (x', y'), after translation, it becomes (x' - 5, y' - 2).
Combining these two transformations, the full transformation that maps triangle P directly to triangle R can be described as:
- Transformation: Rotate triangle P 180º about the origin (0, 0), then translate the resulting triangle by (-5, -2).
Step 2
Write down the coordinates of point A.
Answer
In this problem, point A is stated to be invariant under the transformation that maps triangle P onto triangle R. An invariant point remains unchanged by the transformation.
To find point A, we can analyze the translation component. We represent the transformation from triangle Q to triangle R:
If a point (x, y) is invariant under the translation of (-5, -2), it means:
y = y' - 2
x = x' - 5
Setting these equal, we can find point A:
Let’s make the translation happen in reverse:
- We know: (x, y) = (x' + 5, y' + 2)
- Since point A needs to equal point Q after rotation, we can set it to:
- Assume point Q was at (2.5, 1) after rotation.
After applying the inverse translation:
- Point A = (2.5 + 5, 1 + 2) = (7.5, 3)
Thus, the coordinates of point A are (7.5, 3).