Q1
Answer
For the points $(2,9)$ and $(10,1)$:
  1. Find the distance between them.
  2. Find the midpoint.
  3. Find the gradient between the points.
  4. Find the equation of the line through the points.
  1. $8\sqrt{2}$
  2. $(6,5)$
  3. $-1$
  4. $y=-x+11$
Q2
Answer
The points $A$ and $B$ have coordinates $(0,5)$ and $(2,1)$ respectively.
  1. Find the distance $AB$.
  2. Find the midpoint of $A$ and $B$.
  3. Find the gradient of the line segment $AB$.
  4. Find the equation of the line through the points $A$ and $B$.
  1. $2\sqrt{5}$
  2. $(1,3)$
  3. $-2$
  4. $y=-2x+5$
Q3
Answer
The points $A$, $B$ and $C$ have coordinates $(3,8)$, $(4,3)$ and $(1,6)$ respectively. Show that triangle $ABC$ is right angled.
  1. Find the gradients of $AB$, $AC$ and $BC$.
  2. Find the distances of $AB$, $AC$ and $BC$.
  1. $m_{AB}=-5;\ m_{AC}=1;\ m_{BC}=-1$
  2. $|AB|=\sqrt{26};\ |AC|=\sqrt{8};\ |BC|=\sqrt{18}$ right angle at $C$ by Pythagoras
Q4
Answer
The points $A(1,2)$, $B(5,6)$ and $C(3,8)$ form a triangle.
  1. Find the lengths of all three sides.
  2. Find the midpoint of each side.
  3. Find the gradients of each side.
  4. Determine if the triangle is right-angled.
  1. $|AB|=\sqrt{32};\ |AC|=\sqrt{40};\ |BC|=\sqrt{8}$
  2. $AB: (3,4);\ AC: (2,5);\ BC: (4,7)$
  3. $m_{AB}=1;\ m_{AC}=3;\ m_{BC}=-1$
  4. Yes
Q5
Answer
The points $(2,5)$, $(6,1)$ and $(p,4)$ form a right angled triangle. Find the four possible values of the constant $p$.
$p=1,\; 9,\; 4+\sqrt{7},\; 4-\sqrt{7}$
Q6
Answer
The points $P(2,3)$, $Q(6,7)$ and $R(10,3)$ are three vertices of a parallelogram $PQRS$.
  1. Find the coordinates of $S$.
  2. Show that the opposite sides are parallel.
  1. $(6,-1)$
  2. $m_{PQ}=1,\; m_{QR}=-1,\; m_{RS}=1,\; m_{SP}=-1$
Q7
Answer
The diagonals of a rhombus intersect at right angles and bisect each other. The points $P(1,3)$ and $R(7,5)$ are opposite vertices of a rhombus $PQRS$.
  1. Find the midpoint of $PR$.
  2. Find the equation of the line that goes through $QS$.
  3. Hence find the coordinates of $Q$ and $S$.
  1. $(4,4)$
  2. $y=-3x+16$
  3. $Q=(2,10);\ S=(6,-2)$
Q8
Answer
The points $X(2,2)$, $Y(7,-2)$ and $Z(k,2)$ are vertices of a triangle where $k$ is a constant. Find the two possible values of $k$ for which $XYZ$ is isosceles and $XY = XZ$.
$k = -5;\ 9$