kwizNET Subscribers, please login to turn off the Ads!
Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!

Online Quiz (Worksheet A B C D)

Questions Per Quiz = 2 4 6 8 10

High School Mathematics - 2
9.6 Derivation of distance, external and internal division formulae

Distance Formula
The distance between the points P(x1, y1) and Q(x2, y2) is given by the formula
d = [(x2-x1)2 + (y2-y1)2]
Let P(x1, y1) and Q(x2, y2) be the two points. From P, Q draw PL, QM perpendiculars on the x-axis and PR perpendicular on QM.
Then, PR = LM = x2 - x1
RQ = MQ - MR = MQ - LP = y2 - y1
From right angled triangle PRQ, PQ2 = PR2 + RQ2

d2 = (x2 - x1)2 + (y2 - y1)2
d = 2 - x1)2 + (y2 - y1)2]

Distance from the origin
The distance of the point (x1, y1) from the origin is
d = 1-0)2 + (y1-0)2]
d = 12 + y12]
Example: Find the distance between the points (6, -8), (2, -5).
d2 = (x2-x1)2 + (y2-y1)2
d2 = (2-6)2 + [-5-(-8)]2
d2 = (-4)2 + (3)2
d2 = 16 + 9 = 25
d = 5
Answer: 5 units

Division or section formula
To find the co-ordinates of the point which divides internally the line joining two given points in a given ratio.
Let A(x1, y1) and B(x2, y2) be the two given points and P a point on AB which divides it in the given ratio m1: m2. It is required to find the co-ordinates of P. Suppose they are (x, y). Draw the perpendiculars AL, PM, BN on OX and Ak, PT on PM and BN respectively. Then, from similar triangles APK and PBT we have
AP/PB = AK/PT = KP/TB - 1
Now AP: PB = m1 : m2, AK = LM = OM - OL = x - x1
PT = MN = MK = MP - LA = y - y1
TB = NB - NT = NB - MP = y2 - y

From 1 we have m1/m2 = (x - x1)/(x2 - x) = (y - y1)/(y2 - y)
The first two relations give m1/m2 = (x - x1)/(x2 - x) or m1x2 - m1x = m2x - m2x1
Thus x = (m1x2 + m2x1)/(m1 + m2) , y = (m1y2 + m2y1)/(m1 + m2)
The co-ordinates of mid-point are [(x1+x2)/2, (y1+y2)/2]

Formula for external division
Construct as shown in figure.

From similar triangles APK and PBT,
AP/PB = AK/PT = KP/TB
or m1/m2 = (x - x1)/(x - x2) = (y - y1)/(y - y2)
From the above we get
x = (m1x2 - m2x1)/(m1 - m2)
y = (m1y2 - m2y1)/(m1 - m2)

Example: Find the coordinates of the point which divides the join of the points (8,9) and (-7,4)
a. Internally in the ratio 2:3
b. Externally in the ratio 4:3
Solution:a. The coordinates of the point in the first case
x = (2 x (-7) + 3 x 8)/(2+3) = (-14 + 24)/5 = 2
y = 2x4+3x9/(2+3) = 35/5 = 7
(2,7)
b. For external division the co-ordinates are
x = 4x(-7)-3x8/*4-3) = -52
y = 4x4-3x9/(4-3) = -11
(-52,-11)

Directions: Solve the following problems.

Q 1: Find the coordinates of the point which divides internally the join of the points (1,-2) and (4,7) in the ratio 1:2.
(2,1) or (3,4)
(-5,-4)
(-4,-7)

Q 2: Find the coordinates of the points of trisection of the line joining the points (2,3) and (6,5).
(10/3, 11/3), (14/3, 13/3)
(0,5), (1,5)
(4,8), (7,0)

Q 3: Find the coordinates of the point which divides externally the join of the points (-4,4) and (1,7) in the ratio 2:1.
(6,10)
(5,-6)
(3,7)

Q 4: In what ratio does the point (1,-7/2) divide the join of (-2,-4) and (2,-10/3).
1:6
3:1
1:3

Q 5: Find the coordinates of the point which divides externally the join of the points (3,4) and (-6,2) in the ratio 3:2.
(-6,5)
(8,21)
(21,8)

Q 6: In what ratio is the line joining the points (2,-3) and (5,6) divided by the x-axis.
2:4
1:2
1:7

Q 7: In what ratio is the line joining the points (3,6) and (-6,8) divided by the y-axis.
1:2
1:7
3:7

Q 8: The line joining the points (3,2) and (6,8) is divided into 4 equal parts, find the coordinates of points of section.
(2,5), (6.5), (3,9)
(0,1), (3,6), (6,9)
(15/4,7/2),(9/2,5),(21/4,13/2)

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


Subscription to kwizNET Learning System offers the following benefits:

  • Unrestricted access to grade appropriate lessons, quizzes, & printable worksheets
  • Instant scoring of online quizzes
  • Progress tracking and award certificates to keep your student motivated
  • Unlimited practice with auto-generated 'WIZ MATH' quizzes
  • Child-friendly website with no advertisements
  • Choice of Math, English, Science, & Social Studies Curriculums
  • Excellent value for K-12 and ACT, SAT, & TOEFL Test Preparation
  • Get discount offers by sending an email to discounts@kwiznet.com

Quiz Timer