# Distance and Section Formulae

**Distance formula**

The distance between the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) =**Section formula**

The co-ordinates of the point which divides (internally) the line segment joining the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) in the ratio m_{1}: m_{2}are**Mid-point formula**

The co-ordinates of the mid-point of the line segment joining the points P (x_{1}, y_{1}) and Q_{2}(x_{2}, y_{2}) are**Centroid formula**

The co-ordinates of the centroid of a triangle whose vertices are A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C (x_{3}, y_{3}) 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)