1.03b Straight lines: parallel and perpendicular relationships

5 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.

1. Find the equations of \(A C\) and \(O B\).
2. Find, by calculation, 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 \(A B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium \(A B C D\) in which the coordinates of \(A , B\) and \(C\) are (4, 0), (0, 2) and \(( h , 3 h )\) respectively. The lines \(B C\) and \(A D\) are parallel, angle \(A B C = 90 ^ { \circ }\) and \(C D\) is parallel to the \(x\)-axis.

1. Find, by calculation, the value of \(h\).
2. Hence find the coordinates of \(D\).

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 \(B X\).

9 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_719_958_264_589} The diagram shows a rectangle \(A B C D\), where \(A\) is \(( 3,2 )\) and \(B\) is \(( 1,6 )\).

1. Find the equation of \(B C\). Given that the equation of \(A C\) is \(y = x - 1\), find
2. the coordinates of \(C\),
3. the perimeter of the rectangle \(A B C D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-2_594_778_1360_682} The diagram shows a trapezium \(A B C D\) in which \(B C\) is parallel to \(A D\) and angle \(B C D = 90 ^ { \circ }\). The coordinates of \(A , B\) and \(D\) are \(( 2,0 ) , ( 4,6 )\) and \(( 12,5 )\) respectively.

1. Find the equations of \(B C\) and \(C D\).
2. Calculate the coordinates of \(C\).

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 \(D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_684_771_260_685} The three points \(A ( 1,3 ) , B ( 13,11 )\) and \(C ( 6,15 )\) are shown in the diagram. The perpendicular from \(C\) to \(A B\) meets \(A B\) at the point \(D\). Find

1. the equation of \(C D\),
2. the coordinates of \(D\).

6 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-2_627_748_1685_699} The three points \(A ( 3,8 ) , B ( 6,2 )\) and \(C ( 10,2 )\) are shown in the diagram. The point \(D\) is such that the line \(D A\) is perpendicular to \(A B\) and \(D C\) is parallel to \(A B\). Calculate the coordinates of \(D\).

8 The function f is such that \(\mathrm { f } ( x ) = \frac { 3 } { 2 x + 5 }\) for \(x \in \mathbb { R } , x \neq - 2.5\).

1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and explain why f is a decreasing function.
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. A curve has the equation \(y = \mathrm { f } ( x )\). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

2 Points \(A , B\) and \(C\) have coordinates \(( 2,5 ) , ( 5 , - 1 )\) and \(( 8,6 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find the equation of the line through \(C\) perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\).

9 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-4_767_993_255_575} The diagram shows a quadrilateral \(A B C D\) in which the point \(A\) is ( \(- 1 , - 1\) ), the point \(B\) is ( 3,6 ) and the point \(C\) is (9,4). The diagonals \(A C\) and \(B D\) intersect at \(M\). Angle \(B M A = 90 ^ { \circ }\) and \(B M = M D\). Calculate

1. the coordinates of \(M\) and \(D\),
2. the ratio \(A M : M C\).

11 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-4_885_967_255_589} The diagram shows the curve \(y = ( 6 x + 2 ) ^ { \frac { 1 } { 3 } }\) and the point \(A ( 1,2 )\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).

1. Find the equation of the tangent \(A B\) and the equation of the normal \(A C\).
2. Find the distance \(B C\).
3. Find the coordinates of the point of intersection, \(E\), of \(O A\) and \(B C\), and determine whether \(E\) is the mid-point of \(O A\).

5 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-3_602_751_255_696} The diagram shows a triangle \(A B C\) in which \(A\) has coordinates ( 1,3 ), \(B\) has coordinates ( 5,11 ) and angle \(A B C\) is \(90 ^ { \circ }\). The point \(X ( 4,4 )\) lies on \(A C\). Find

1. the equation of \(B C\),
2. the coordinates of \(C\).

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 \(C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle \(A B C D\) in which point \(A\) is ( 0,8 ) and point \(B\) is ( 4,0 ). The diagonal \(A C\) has equation \(8 y + x = 64\). Find, by calculation, the coordinates of \(C\) and \(D\).

3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).

1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
2. Find the distance \(A C\).

6 Points \(A , B\) and \(C\) have coordinates \(A ( - 3,7 ) , B ( 5,1 )\) and \(C ( - 1 , k )\), where \(k\) is a constant.

1. Given that \(A B = B C\), calculate the possible values of \(k\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at \(D\).
2. Calculate the coordinates of \(D\).

3 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-2_515_750_669_699} In the diagram \(O C A\) and \(O D B\) are radii of a circle with centre \(O\) and radius \(2 r \mathrm {~cm}\). Angle \(A O B = \alpha\) radians. \(C D\) and \(A B\) are arcs of circles with centre \(O\) and radii \(r \mathrm {~cm}\) and \(2 r \mathrm {~cm}\) respectively. The perimeter of the shaded region \(A B D C\) is \(4.4 r \mathrm {~cm}\).

1. Find the value of \(\alpha\).
2. It is given that the area of the shaded region is \(30 \mathrm {~cm} ^ { 2 }\). Find the value of \(r\). \(4 C\) is the mid-point of the line joining \(A ( 14 , - 7 )\) to \(B ( - 6,3 )\). The line through \(C\) perpendicular to \(A B\) crosses the \(y\)-axis at \(D\).

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.

11 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.

1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.

3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(A B\).
2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).

4 Two points \(A\) and \(B\) have coordinates \(( - 1,1 )\) and \(( 3,4 )\) respectively. The line \(B C\) is perpendicular to \(A B\) and intersects the \(x\)-axis at \(C\).

1. Find the equation of \(B C\) and the \(x\)-coordinate of \(C\).
2. Find the distance \(A C\), giving your answer correct to 3 decimal places.