(a) Using a scale of 2cm to 2units on both axes, draw on a sheet of graph paper two perpendicular axes 0x and 0y for \(-10 \leq x \leq 10\) and \(-10 \leq y...
Question 1 Report
(a) Using a scale of 2cm to 2units on both axes, draw on a sheet of graph paper two perpendicular axes 0x and 0y for \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 10\)
(b) Given the points P(3, 2). Q(-1. 5). R(0. 8) and S(3, 7). draw on the same graph, indicating clearly the vertices and their coordinates, the:
(i) quadrilateral PQRS;
(ii) image \(P_1Q_1R_1S_1\) of PQRS under an anticlockwise rotation of \(90^o\) about the origin where \(P \to P_1\), \(Q \to Q_1\), \(R \to R_1\) and \(S \to S_1\)
(iii) image \(P_2Q_2R_2S_2\) of \(P_1Q_1R_1S_1\) under a reflection in the line \(y - x = 0\) where \(P_1 \to P_2\), \(Q_1 \to Q_2\), \(R_1 \to R_2\) and \(S_1 \to S_2\)
(c) Describe precisely the single transformation T for which \(T : PQRS \to P_2Q_2R_2S_2\)
(d) The side \(P_1Q_1\) of the quadrilateral \(P_1Q_1R_1S_1\) cuts the x-axis at the point W. What type of quadrilateral is \(P_1S_1R_1W\)?
Answer Details
(a) and (b) The required coordinate graph, drawn to equal scales on both axes, is shown below. The vertices are joined in the order in which they are named.
The coordinates plotted are:
| Object | Vertices |
|---|---|
| PQRS | P(3,2), Q(-1,5), R(0,8), S(3,7) |
| P1Q1R1S1 | P1(-2,3), Q1(-5,-1), R1(-8,0), S1(-7,3) |
| P2Q2R2S2 | P2(3,-2), Q2(-1,-5), R2(0,-8), S2(3,-7) |
For a rotation of \(90^\circ\) anticlockwise about the origin, \((x,y)\mapsto(-y,x)\). Thus, for example, \(P(3,2)\mapsto P_1(-2,3)\).
Reflection in \(y=x\) maps \((x,y)\mapsto(y,x)\). Thus \(P_1(-2,3)\mapsto P_2(3,-2)\).
(c) Combining the two transformations gives
\[(x,y)\mapsto(-y,x)\mapsto(x,-y).\]
Hence \(T\) is a reflection in the x-axis.
(d) The line \(P_1Q_1\) has gradient
\[m=\frac{-1-3}{-5-(-2)}=\frac{-4}{-3}=\frac{4}{3}.\]
Its equation is
\[y-3=\frac{4}{3}(x+2).\]
At the x-axis, \(y=0\), so
\[-3=\frac{4}{3}(x+2),\qquad x=-\frac{17}{4}.\]
Therefore \(W\left(-\frac{17}{4},0\right)\). Since \(P_1S_1\) and \(R_1W\) are both horizontal, they are parallel. Thus \(P_1S_1R_1W\) is a trapezium.
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