Perpendicular bisector of segment

9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).

1. Find the equation of \(M B\) and the coordinates of \(B\).
2. Show that \(A B\) is perpendicular to \(B C\).
3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).

1 Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to \(( 10,3 )\) meets the \(x\)-axis.

8 A curve has equation \(y = 3 x - \frac { 4 } { x }\) and passes through the points \(A ( 1 , - 1 )\) and \(B ( 4,11 )\). At each of the points \(C\) and \(D\) on the curve, the tangent is parallel to \(A B\). Find the equation of the perpendicular bisector of \(C D\).

2 The point \(A\) has coordinates ( \(- 2,6\) ). The equation of the perpendicular bisector of the line \(A B\) is \(2 y = 3 x + 5\).

1. Find the equation of \(A B\).
2. Find the coordinates of \(B\).

8 Points \(A\) and \(B\) have coordinates \(( h , h )\) and \(( 4 h + 6,5 h )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = k\). Find the values of the constants \(h\) and \(k\).

6 The coordinates of points \(A\) and \(B\) are \(( - 3 k - 1 , k + 3 )\) and \(( k + 3,3 k + 5 )\) respectively, where \(k\) is a constant ( \(k \neq - 1\) ).

1. Find and simplify the gradient of \(A B\), showing that it is independent of \(k\).
2. Find and simplify the equation of the perpendicular bisector of \(A B\).

2 Two points \(A\) and \(B\) have coordinates \(( 1,3 )\) and \(( 9 , - 1 )\) respectively. The perpendicular bisector of \(A B\) intersects the \(y\)-axis at the point \(C\). Find the coordinates of \(C\).

5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).

1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
2. Find, by calculation, the distance \(B X\).

7 Three points have coordinates \(A ( 2,6 ) , B ( 8,10 )\) and \(C ( 6,0 )\). The perpendicular bisector of \(A B\) meets the line \(B C\) at \(D\). Find

1. the equation of the perpendicular bisector of \(A B\) in the form \(a x + b y = c\),
2. the coordinates of \(D\).

7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).

1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).

6 The points \(A ( 1,1 )\) and \(B ( 5,9 )\) lie on the curve \(6 y = 5 x ^ { 2 } - 18 x + 19\).

1. Show that the equation of the perpendicular bisector of \(A B\) is \(2 y = 13 - x\).
   The perpendicular bisector of \(A B\) meets the curve at \(C\) and \(D\).
2. Find, by calculation, the distance \(C D\), giving your answer in the form \(\sqrt { } \left( \frac { p } { q } \right)\), where \(p\) and \(q\) are integers.

6. The point \(A\) has coordinates \(( - 4,11 )\) and the point \(B\) has coordinates \(( 8,2 )\).

1. Find the gradient of the line \(A B\), giving your answer as a fully simplified fraction. The point \(M\) is the midpoint of \(A B\). The line \(l\) passes through \(M\) and is perpendicular to \(A B\).
2. Find an equation for \(l\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found. The point \(C\) lies on \(l\) such that the area of triangle \(A B C\) is 37.5 square units.
3. Find the two possible pairs of coordinates of point \(C\).

   VIXV SIHIANI III IM IONOO

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1. The line \(l _ { 1 }\) has equation \(x + 3 y - 11 = 0\)

The point \(A\) and the point \(B\) lie on \(l _ { 1 }\) Given that \(A\) has coordinates ( \(- 1 , p\) ) and \(B\) has coordinates ( \(q , 2\) ), where \(p\) and \(q\) are integers,

1. find the value of \(p\) and the value of \(q\),
2. find the length of \(A B\), giving your answer as a simplified surd. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\).
3. Find an equation for \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

The points \(P\) and \(Q\) have coordinates \(( - 1,6 )\) and \(( 9,0 )\) respectively. The line \(l\) is perpendicular to \(P Q\) and passes through the mid-point of \(P Q\).
Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

10 \begin{figure}[h] \end{figure} In Fig.10, A has coordinates \(( 1,1 )\) and C has coordinates \(( 3,5 )\). M is the mid-point of AC . The line \(l\) is perpendicular to AC.

1. Find the coordinates of M . Hence find the equation of \(l\).
2. The point B has coordinates \(( - 2,5 )\). Show that B lies on the line \(l\).
   Find the coordinates of the point D such that ABCD is a rhombus.
3. Find the lengths MC and MB . Hence calculate the area of the rhombus ABCD .

1 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks. \(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.

1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12.
   [diagram]

5 The points \(A\) and \(B\) have coordinates \(( 2,1 )\) and \(( 5 , - 3 )\) respectively.

1. Find the length of \(A B\).
2. Find an equation of the line through the mid-point of \(A B\) which is perpendicular to \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks. \(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.

1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_451_483_776_790} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Not to scale

The coordinates of points \(A\) and \(B\) are \((-3k - 1, k + 3)\) and \((k + 3, 3k + 5)\) respectively, where \(k\) is a constant \((k \neq -1)\).

1. Find and simplify the gradient of \(AB\), showing that it is independent of \(k\). [2]
2. Find and simplify the equation of the perpendicular bisector of \(AB\). [5]

\(A\) is the point \((-2, 6)\) and \(B\) is the point \((3, -8)\). The line \(l\) is perpendicular to the line \(x - 3y + 15 = 0\) and passes through the mid-point of \(AB\). Find the equation of \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

The point \(A\) has coordinates \((-1, a)\) and the point \(B\) has coordinates \((3, b)\) The line \(AB\) has equation \(5x + 4y = 17\) Find the equation of the perpendicular bisector of the points \(A\) and \(B\). [4 marks]