The coordinates of points \(A, B, C\) and \(D\) are as follows. \(A(-4,3) \quad B(6,-9) \quad C(15,10) \quad D(14,-1)\)

The line \(L\) has equation \(y=11 x-75\). The perpendicular bisector of the line \(A B\) meets \(L\) at the point \(E\). Find the area of triangle \(C D E\).

The point \(A\) has coordinates \((1,4)\) and the point \(B\) has coordinates \((5,6)\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at the point \(C\) and the \(y\)-axis at the point \(D\). Given that \(O\) is the origin, find the area of triangle \(O C D\).

The points \(A\) and \(B\) have coordinates \((2,5)\) and \((10,-15)\) respectively. The point \(P\) lies on the perpendicular bisector of \(AB\). The \(y\)-coordinate of \(P\) is \(-9\).

(a) Find the coordinates of \(P\).

(b) The point \(R\) is the reflection of \(P\) in the line \(AB\). Find the coordinates of \(R\).

The straight line \(y=3x-11\) and the curve \(xy=4-3x-2x^2\) intersect at the points \(A\) and \(B\). The point \(C\), with coordinates \((a,-8)\), where \(a\) is a constant, lies on the perpendicular bisector of the line \(AB\). Find the value of \(a\).

The perpendicular bisector of the line joining the points \((-3,\frac23)\) and \((6,-\frac73)\) passes through the point \((2,k)\). Find the value of \(k\).

A line, \(L\), has equation \(4x+5y=9\). Points \(A\) and \(B\) have coordinates \((-6,7)\) and \((1,9)\), respectively. Find the equation of the line parallel to \(L\) which passes through the midpoint of \(AB\).

The points \(A\), \(B\) and \(C\) have coordinates \((2,6)\), \((6,1)\) and \((p,q)\), respectively. The point \(B\) is the midpoint of \(AC\). Find the equation of the line through \(C\) that is perpendicular to \(AB\), giving your answer in the form \(ax+by=c\), where \(a\), \(b\) and \(c\) are integers.

The line \(AB\) is such that the points \(A\) and \(B\) have coordinates \((-4,6)\) and \((2,14)\) respectively.

(a) The point \(C\), with coordinates \((7,a)\), lies on the perpendicular bisector of \(AB\). Find the value of \(a\).

(b) Given that the point \(D\) also lies on the perpendicular bisector of \(AB\), find the coordinates of \(D\) such that the line \(AB\) bisects the line \(CD\).

The line

\(x+2y=10\)

intersects the two lines satisfying the equation

\(|x+y|=2\)

at the points \(A\) and \(B\).

(a) Show that the point \(C(-5,20)\) lies on the perpendicular bisector of the line \(AB\).

(b) The point \(D\) also lies on this perpendicular bisector. \(M\) is the mid-point of \(AB\). The distance \(CD\) is three times the distance \(CM\). Find the possible coordinates of \(D\).

The points \(A(5,-4)\) and \(C(11,6)\) are such that \(AC\) is the diagonal of a square, \(ABCD\).

(a) Find the length of the line \(AC\).

(b)

(i) The coordinates of the centre, \(E\), of the square are \((8,y)\). Find the value of \(y\).

(ii) Find the equation of the diagonal \(BD\).

(iii) Given that the \(x\)-coordinate of \(B\) is less than the \(x\)-coordinate of \(D\), write \(\overrightarrow{EB}\) and \(\overrightarrow{ED}\) as column vectors.

The curves \(y=x^2+x-1\) and \(2y=x^2+6x-2\) intersect at the points \(A\) and \(B\).

(a) Show that the mid-point of the line \(AB\) is \((2,9)\).

The line \(l\) is the perpendicular bisector of \(AB\).

(b) Show that the point \(C(12,7)\) lies on the line \(l\).

(c) The point \(D\) also lies on \(l\), such that the distance of \(D\) from \(AB\) is two times the distance of \(C\) from \(AB\). Find the coordinates of the two possible positions of \(D\).

Solutions by accurate drawing will not be accepted.

The points \(A\) and \(B\) have coordinates \((-2,4)\) and \((6,10)\) respectively.

(a) Find the equation of the perpendicular bisector of the line \(AB\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

The point \(C\) has coordinates \((5,p)\) and lies on the perpendicular bisector of \(AB\).

(b) Find the value of \(p\).

It is given that the line \(AB\) bisects the line \(CD\).

(c) Find the coordinates of \(D\).

The line \(y=5x+6\) meets the curve \(xy=8\) at the points \(A\) and \(B\).

(a) Find the coordinates of \(A\) and of \(B\).

(b) Find the coordinates of the point where the perpendicular bisector of the line \(AB\) meets the line \(y=x\).

Solutions to this question by accurate drawing will not be accepted.

The points \(A\) and \(B\) are \((4,3)\) and \((12,-7)\) respectively.

(a) Find the equation of the line \(L\), the perpendicular bisector of the line \(AB\).

(b) The line parallel to \(AB\) which passes through the point \((5,12)\) intersects \(L\) at the point \(C\). Find the coordinates of \(C\).

Solutions to this question by accurate drawing will not be accepted.

Find the equation of the perpendicular bisector of the line joining the points \((4,-7)\) and \((-8,9)\).

(a) Find the equation of the perpendicular bisector of the line joining the points \((12,1)\) and \((4,3)\), giving your answer in the form \(y=mx+c\).

(b) The perpendicular bisector cuts the axes at points \(A\) and \(B\). Find the length of \(AB\).

The points \(A\) and \(B\) have coordinates \((p,3)\) and \((1,4)\) respectively, and the line \(L\) has equation \(3x+y=2\).

(i) Given that the gradient of \(AB\) is \(\frac13\), find \(p\).

(ii) Show that \(L\) is the perpendicular bisector of \(AB\).

(iii) Given that \(C(q,-10)\) lies on \(L\), find \(q\).

(iv) Find the area of triangle \(ABC\).

Solutions to this question by accurate drawing will not be accepted.

\(P\) is the point \((8,2)\) and \(Q\) is the point \((11,6)\).

(i) Find the equation of the line \(L\), which passes through \(P\) and is perpendicular to the line \(PQ\).

The point \(R\) lies on \(L\) such that the area of triangle \(PQR\) is \(12.5\) units\(^2\).

(ii) Showing all your working, find the coordinates of each of the two possible positions of point \(R\).

Find the equation of the perpendicular bisector of the line joining the points \((1,3)\) and \((4,-5)\). Give your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

Find the equation of the perpendicular bisector of the line joining the points \((1,3)\) and \((4,-5)\). Give your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.