Coordinate Geometry

Coordinate Geometry

4.1 Distance/length of a line segment

Find the distance between each of the following:

1. $(2,1),(5,5)$

2. $(3,6),(8,18)$

3. $(-3,2),(5,8)$

4. The three points $A$,$B$ and $C$ have coordinates $(-1,3)$, $(6,4)$ and $(1,-1)$ respectively. Show that $AB=BC$

5. $P(-2,-6)$,$Q(6,9)$ and $R(1,-3)$ are the vertices of a triangle. Find the number of units by which the length of $PQ$ exceeds the length of $QR$

6. Prove that $A(2,3)$,$B(5,6)$ and $C(8,3)$ are the vertices of a right-angled triangle

7. A triangle has vertices at $(1,2)$,$(13,7)$ and $(6,14)$. Prove that the triangle is isosceles.

8. Coordinates of the points $C$,$D$ and $E$ are $(1,3)$,$(5,-1)$ and $(-1,-3)$ respectively. Prove that the triangle $CDE$ is isosceles

9. Showing that $A(-3,-1)$,$B(1,-4)$ and $C(7,4)$ form a right-angled triangle, deduced that the area of $\Delta ABC$ is $25cm^{2}$

10. The points $L,M$ and $N$ have coordinates $(3,1)$,$(2,6)$ and $(x,5)$ respectively. Given that the distance $LM$ is equal to the distance $MN$, calculate the possible values of $x$.

4.2 Gradient of a line segment

Find the gradient of the straight line joining each of the following pairs of points

1. $(2,3),(4,7)$

2. $(-1,2),(1,8)$

3. $(5,4),(3,3)$

4. Given $A(2,3)$,$B(5,5)$,$C(7,2)$ and $D(4,0)$

     a) Prove that $AB$ is parallel to $DC$

     b) Prove that $AC$ is perpendicular to $BD$

5.  Show that the lines joining the point $A(2,5)$ to $B(5,12)$ and $C(-2,-6)$ to $D(1,1)$ are parallel

6. A triangle has vertices $A(3,-2)$,$B(2,-14)$ and $C(-2,-4)$. Find the gradients of the straight lines $AB$, $BC$ and $CA$. Hence prove that the triangle is right-angled.

7. The straight line joining the point $A(a,3)$ to the point $B(5,7)$ is parallel to the straight line joining the point $B$ to the point $C(-3,-1)$. Calculate the value of $a$.

8. The straight line joining the point $P(5,6)$ to the point $Q(q,2)$ is perpendicular to the straight line joining the point $Q$ to the point $R(9,-1)$. Calculate the possible values of $q$.

9. A quadrilateral has vertices $A(-2,-3)$,$B(1,-1)$,$C(7,-10)$ and $D(2,-9)$. Prove that

     a) $AD$ is parallel to $BC$

     b) $AB$ is perpendicular to $BC$

     c) the area of the quadrilateral $ABCD$ is $32\frac{1}{2}unit^{2}$

4.3 Mid-point

Find the coordinates of the mid-point of the straight line joining each of the following pairs of points

1. $X(3,2),Y(7,4)$

2. $P(3,5),Q(7,7)$

3. $R(-2,3),S(4,1)$

4. $M(6,5)$ is the mid-point of the straight line joining $A(2,3)$ to the point $B$. Find the coordinates of $B$

5. Find the coordinates of the point $S$, given that $M(3,-2)$ is the mid-point of the straight line joining $S$ to $T(9,-2)$

6. $P$ is the mid-point of the straight line joining $C(-5,3)$ to the point $D$. Given that $P$ has coordinates $(2,1)$, find the coordinates of $D$

7. Prove that the points $A(1,-1)$,$B(2,-5)$ and $C(-2,-4)$ are the vertices of an isosceles triangle. Given that $M$ is the mid-point of the longest side of the triangle $ABC$, calculate the coordinates of $M$.

4.4 Equation of straight line

Find the gradient of each of the following straight lines.

1. $y=5x-2$

2. $y=3x-5$

3. $5+3x+2y=0$

4. $\frac{y}{3}+\frac{x}{5}=4$

Find the equation of the straight line that has the following properties

5. Gradients $2$ and passes through $(5,3)$

6. Gradients $-2$ and passes through $(6,-3)$

7. Gradients $\frac{1}{4}$ and passes through $(2,5)$

8. Gradients $-\frac{2}{3}$ and passes through $(-4,2)$

9. Passes through $(4,3)$ and is parallel to $y=3x+5$

10. Passes through $(4,-1)$ and is parallel to $y=3-2x$

11. Passes through $(6,-2)$ and is perpendicular to $y=-3x+4$

12. Passes through $(-6,3)$ and is perpendicular to $4y+3x=8$

4.5 Equation of perpendicular bisector to a straight line

Find the equation of the perpendicular bisector of the straight line joining each of the  following pairs of points

1. $(2,-5)$ and $(4,-1)$

2. $(5,4)$ and $(2,-2)$

3. Find the equation of the straight line, $p$, which is the perpendicular bisector of the straight line joining the points $(1,2)$ and $(5,4)$. The line $p$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Calculate the area of the triangle $OAB$

4. The perpendicular bisector of the straight line joining the points $(3,2)$ and $(5,6)$ meets the $x$-axis at $A$ and the $y$-axis at $B$. Prove that the distance $AB$ is equal to $6\sqrt{5}$

4.6 Points of intersection

Find the coordinates of the point of intersection of each of the following pairs of straight lines

1. $2y+3x=1,3y+x=5$

2. $5y-3x=1,2y+x=7$

3. $x+3y-2=0,3x+5y-8=0$

4. $2x+5y+6=0,3x+4y-2=0$

5. a. Find the coordinates of the points of intersection, $P$, of the lines $2x+y=7$ and $3x-4y=5$

b. Show that the lines $x+2y=5$ also passes through the point $P$

6. Show that the three lines $2x+3y=4$, $x-2y=9$ and $3x+7y=1$ are concurrent.

7. Find the equation of the line which is parallel to the line $y-3x+5=0$, and which is passes through the point of intersection of the lines $2y+3x-5=0$ and $3y-2x=14$

8. a. Find the equation of the straight line, $l$, which passes through the point $(2,4)$ and which is perpendicular to the line $5y+x=7$

b. Given that the line $l$ meets the line $y=x+6$ at the point $S$, find the coordinates of the point $S$

9. a. The line $y=2x$ meets the line $x+3y=14$ at the point $A$, find the coordinates of $A$

b. The line $x+3y=14$ meets $x$-axis at $B$, find the coordinates of $B$

c. Calculate the area of the triangle $OAB$ when O is the origin

10. Find the coordinates of the vertices of the triangle which has sides given by the equations $x+y=3$, $x-2y=1$ and $3x=2y$

11. Three points have coordinates $L(2,5)$, $M(-2,3)$ and $N(4,9)$. Find

     a. the equation of the perpendicular bisector, $p$, of $LM$

     b. the coordinates of the point where $p$ meets $MN$

12. Prove that the triangle which has sides given by the equations $3y+x=8$, $y+3x=24$ and $y=x$ is isosceles.

4.7 More exercise

1. The line $x+2y=9$ intersects the curve $xy+18=0$ at the points $A$ and $B$. Find the coordinates of $A$ and $B$

2. The diagram shows a rectangle $ABCD$, where $A$ is $(3,2)$ and $B$ is $(1,6)$

     i) Find the equation of $BC$

     Given that the equation of $AC$ is $y=x-1$, find

     ii) the coordinates of $C$

     iii) the perimeter of the rectangle $ABCD$

3. The line $L_1$ has equation $2x+y=8$. The line $L_2$ passes through the point $A(7,4)$ and is perpendicular to $L_1$

     i) Find the equation of $L_2$

     ii) Given that the lines $L_1$ and $L_2$ intersect at the point $B$, find the length of $AB$

4. Find the coordinates of the points of intersection of the line $y+2x=11$ and the curve $xy=12$

5. The curve $y=9-\frac{6}{x}$ and the line $y+x=8$ intersect at two points. Find

     i) the coordinates of the two points

     ii) The equation of the perpendicular bisector of the line joining the two points.

6. The equation of the curve is $y=x^{2}-4x+7$ and the equation of the line is $y+3x=9$. The curve and the line intersect at the point $A$ and $B$

     i) The mid-point of $AB$ is $M$. Show that the coordinates of $M$ are $(\frac{1}{2},7\frac{1}{2})$

     ii) Find the coordinates of the point $Q$ on the curve at which the tangent is parallel to the line $y+3x=9$

     iii) Find the distance $MQ$

7. The diagram shows a rhombus $ABCD$. The points $B$ and $D$ have the coordinates $(2,10)$ and $(6,2)$ respectively, and $A$ lies on the $x$-axis. The mid-point of $BD$ is $M$. Find, by calculation, the coordinates of each of $M$ and $A$ and $C$

8. The curve $y^{2}=12x$ intersects the line $3y=4x+6$ at two points. Find the distance between the two points.

9. Find the equation of the straight line, $p_1$, which is the perpendicular bisector of the points $A(-2,3)$ and $B(1,-5)$ and the equation of straight line $p_2$, which is the perpendicular bisector of the points $B(1,-5)$ and $C(17,1)$. Show that $p_1$ is perpendicular to $p_2$