Parallel and perpendicular lines

PQRS is a square.

PR is a diagonal of the square.

P is the point with coordinates (4, 7) R is the point with coordinates (8, –5)

Find an equation of the straight line that passes through the points Q and S Give your answer in the form ay = bx + c where a, b and c are integers.

ABCD is a kite.

AB = AD and CB = CD

The point B has coordinates (k, 1) where k is a negative constant. The point D has coordinates (8, 7)

The straight line L passes through the points B and D

The straight line L is parallel to the line with equation 5y – 3x = 6

Find an equation of AC Give your answer in the form px + qy = r where p, q and r are integers. Show your working clearly.

G is the point on the curve with equation y = 8x² – 14x – 6 where the gradient is 10 The straight line Q passes through the point G and is perpendicular to the tangent at G

Find an equation for Q Give your answer in the form ax + by + c = 0 where a, b and c are integers.

ABCD is a trapezium with AB parallel to DC

A is the point with coordinates (–4, 6) B is the point with coordinates (2, 3) D is the point with coordinates (–1, 8)

The trapezium has one line of symmetry. The line of symmetry intersects CD at the point E

Work out the coordinates of the point E

The centre O of a circle has coordinates (4, 7)

The point A, on the circle, has coordinates (6, 11) and AOP is a diameter of the circle.

Find an equation of the tangent to the circle at the point P

ABC is a triangle in which angle ABC = 90°

p and q are integers such that

the coordinates of A are (p, 10) the coordinates of B are (–1, –5) the coordinates of C are (8, q)

Given that the gradient of AC is –

work out the value of p and the value of q

The straight line L passes through point A (– 6, 2) and point B (5, 3) The straight line M is perpendicular to L and passes through the midpoint of A and B. The line M intersects the line x = – 1 at point C.

Calculate the area of triangle ABC.

Point A has coordinates (5, 8) Point B has coordinates (9, –4)

Work out the gradient of AB.

The straight line L has equation y = –4x + 5

Write down the gradient of a straight line that is perpendicular to L.

ABC is an isosceles triangle with AB = AC.

B is the point with coordinates (–1, 5) C is the point with coordinates (2, 10) M is the midpoint of BC.

Find an equation of the line through the points A and M. Give your answer in the form py + qx = r where p, q and r are integers.

ABCD is a rhombus.

The diagonals, AC and BD, intersect at the point M. The coordinates of M are (6, −11)

The points A and C both lie on the line with equation 2y + 7x = 20

Find the exact coordinates of the point where the line through B and D intersects the y-axis.

Use the graph to find an estimate for the gradient of the curve at P.

Hence find an equation of the tangent to the curve at P. Give your answer in the form y=mx+c

The equation f (x) =k has exactly two different solutions for −4≤ x ≤12

Use the graph to find the two possible values of k.

L₁ and L₂ are two straight lines. The origin of the coordinate axes is O.

L₁ has equation 5x + 10y = 8 L₂ is perpendicular to L₁ and passes through the point with coordinates (8, 6) L₂ crosses the x-axis at the point A. L₂ intersects the straight line with equation x = –3 at the point B. Find the area of triangle AOB. Show your working clearly.

The straight line L₁ has equation 2y = 6x

The straight line L₂ is perpendicular to L₁ and passes through the point (9, í1)

Find an equation for L₂ Give your answer in the form ay + bx = c

ABCD is a kite with AB = AD and CB = CD.

B is the point with coordinates (10, 19) D is the point with coordinates (2, 7)

Find an equation of the line AC. Give your answer in the form py + qx = U where p, q and U are integers.

ABC is an isosceles triangle such that

AB = AC A has coordinates (4, 37) B and C lie on the line with equation 3 y = 2 x + 12

Find an equation of the line of symmetry of triangle ABC. Give your answer in the form px + qy = U where p, q and U are integers. Show clear algebraic working.

The straight line L₁ has equation y = 6 – 2x The straight line L₂ is perpendicular to L₁ and passes through the point (4, 7)

Find the coordinates of the point where the line L₂ crosses the x-axis.