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Home » Articles » LESSON NOTES (Mathematics) » High School level


DISTANCE BETWEEN TWO POINTS

DISTANCE BETWEEN TWO POINTS

 

Consider the figure below:

 

Let P and Q be points with coordinates (x1,y1) and (x2,y2) respectively as shown above.

PQR is right angled triangle. By Pythagoras theorem:

PQ2 = PR2 + RQ2 = (x2-x1)2 + (y2 – y1)2

 

Therefore:  PQ2 =  (x2-x1)2 + (y2 – y1)2

 

Example #1

Find the distance between points (5,-2) and (2,2)

Solution:

From the above points:

x1 = 5, y1 = -2   and  x2 = 2, y2 = 2  

Therefore by distance formula:

PQ2 =  (x2-x1)2 + (y2 – y1)2

Distance =√[ (5 – 2)2 + (-2 -2)2]

Distance =√[ 32 + (-4)2]  = √ (9 + 16) = √[ 25]

Therefore, the distance = 5 units

 

Example #2

Prove that the Triangle with vertices given by A(3,5), B(-1,-1) and C(4,4) is a right angled triangle.

Solution:

AB2 = (-1 - 3)2 + (-1 - 5)2

AB2  = 16 + 36 = 52

 

BC2 = (-1 - 4)2 + (-1 - 4)2

BC2  = 25 + 25 = 50

 

AC2 = (4 - 3)2 + (4 - 5)2

AC2  = 1 + 1 = 2

 

Relating the three sides we found that:

AB2 = BC2 +  AC2

Hence Triangle ABC is the right angled triangle.

 

Try yourself:

 

1 .Find the distance between each of the following points:

                                (a)  A(6,2)   and  B(-2,4)

                                (b)  C(-2,2)  and  D(8,-2)

                                (c)  E(3,1)    and   F(-2,6)

                                (d)  G(3,7)   and   H(9,-2)

 2. Find the perimeter of triangle given by vertices (2,1) , (6,1) and (6,4)

 3. Find the area of the Triangle with vertices given by A(3,5), B(-1,-1) and C(4,4)

  4. Given points: A(3,1), B(0,6) and C(-5,3). Prove that triangle PQR is isosceles.

 

********Thanks for reading and have a nice day******

Added By (Yahyou M) - BADSHAH

 

Category: High School level | Added by: Admin (02/Sep/2016) | Author: Yahya Mohamed E W
Views: 401 | Tags: Plane, Distance, Points, coordinate geometry | Rating: 0.0/0
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