Distance and Section Formulae
- Distance formula
The distance between the points P(x1, y1) and Q(x2,
y2) =
- Section formula
The co-ordinates of the point which divides (internally) the line segment
joining the points P(x1, y1) and Q(x2, y2)
in the ratio m1 : m2 are
- Mid-point formula
The co-ordinates of the mid-point of the line segment joining the points P (x1,
y1) and Q2 (x2, y2) are
- Centroid formula
The co-ordinates of the centroid of a triangle whose vertices are A (x1,
y1), B (x2, y2) and C (x3, y3)
are
Exercise
- Calculate the distance between the points P(2, 2), Q(5, 4) correct to
three significant figures. (Do not consult tables).
- A is a point on the y-axis whose ordinate is 5 and B is the point (-3, 1).
Calculate the length of AB.
- The distance between A(1, 3) and B(x, 7) is 5. Find the possible values of x.
- P and Q have co-ordinates (-1, 2) and (6, 3) respectively. Reflect P in
the x-axis to P'. Find the length of the segment P'Q.
- Point A(2, -4) is reflected in the origin as A'. Point B(-3, 2) is
reflected in x-axis at B'. Write the co-ordinates of A' and B'. Calculate the
distance A'B' correct to one decimal place.
- The center of a circle of radius 13 units is the point (3, 6). P(7, 9) is
a point inside the circle. APB is a chord of the circle such that AP = PB.
Calculate the length of AB.
- A and B have co-ordinates (4, 3) and (0, 1) respectively. Find
(i) the image A' of A under reflection in the y-axis.
(ii) the image B' of B under reflection in the line AA'.
(iii) the length of A'B'.
- What point (or points) on the x-axis are at a distance of 5 units from the point (5, -4)?
- Find point (or points) which are at a distance of
10 from the point (4, 3), given
that the ordinate of the point (or points) is twice the abscissa.
- Show that the points (3, 3), (9, 0) and (12, 21) are the vertices of a
right angled triangle.
- Show that the points (0, -1), (-2, 3), (6, 7) and (8, 3) are the vertices of a rectangle.
- The points A(0, 3), B(-2, a) and C(-1, 4) are the vertices of a right
angled triangle at A, find the value of a.
- Show by distance formula that the points (-1, -1), (2, 3) and (8, 11) are collinear.
- Calculate the co-ordinates of the point P that divides the line joining
the points A (-1, 3) and B(5, -6) internally in the ratio 1:2.
- Find the co-ordinates of the points of trisection of the line segment
joining the points (3, -3) and (6, 9).
- The line segment joining A(-3, 1) and B(5, -4) is a diameter of a circle
whose center is C. Find the co-ordinates of the point C.
- The mid-point of the line joining (a, 2) and (3, 6) is (2, b). Find the values of a and b.
- The mid-point of the line segment joining (2a, 4) and (-2, 3b) is (1, 2a
+1). Find the values of a and b.
- The center of a circle is (1, -2) and one end of a diameter is (-3, 2),
find the co-ordinates of the other end.
- Find the reflection of the point (5, -3) in the point (-1, 3).
Answers
1. 3·61 units
2. 5units
3. 4 or -2
4. 74 units
5. A'(-2, 4), B'(-3, -2); 6·1 units
6. 24 units
7. (i) (-4, 3)
(ii) (0, 5)
(iii) 25 units
8. (2, 0) and (8, 0 .
9. (1, 2), (3, 6)
10. 67·5 sq. units
12. 1
14. (1, 0)
15. (4, 1), (5, 5)
16. (1,-3/2)
17. a = 1, b = 4
18. a = 2, b = 2
19. (5, -6)
20. (-7, 9)