The gradient formula
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The Gradient Formula. Gradient of AB = m AB = height / horizontal = BC / AC = (y 2 -y 1 ) (x 2 -x 1 ). B(x 2 ,y 2 ). y 2 -y 1. A(x 1 ,y 1 ). x 2 -x 1. C(x 2 ,y 1 ). The Gradient Formula ctd. Summary If A is (x 1 ,y 1 ) and B is (x 2 ,y 2 )

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The Gradient Formula

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The gradient formula

The Gradient Formula

  • Gradient of AB

  • = mAB

  • = height/horizontal

  • = BC/AC

  • = (y2-y1) (x2-x1)

B(x2,y2)

y2-y1

A(x1,y1)

x2-x1

C(x2,y1)


The gradient formula ctd

The Gradient Formula ctd.

  • Summary

  • If A is (x1,y1) and B is (x2,y2)

  • then the gradient of AB is given by

  • mAB = (y2-y1) (x2-x1)


Example 4

Example 4

P is (5, 9) and Q is (7,17).

  • mPQ = (y2-y1) (x2-x1)

  • = (17 - 9)

  • ( 7 - 5)

  • = 8/2

  • = 4

NB: line looks like

Q

P


Example 5

Example 5

E is (15, 9) and F is (7,17).

  • mEF = (y2-y1) (x2-x1)

  • = (17 - 9)

  • ( 7 - 15)

  • = 8/-8

  • = -1

NB: line looks like

F

E


Example 6

Example 6

V is (-5, -9) and W is (12,-9).

  • mVW = (y2-y1) (x2-x1)

  • = (-9 -(- 9))

  • ( 12 - (-5))

  • = 0/17

  • = 0

  • NB: A horizontal line has no steepness.

NB: line looks like

V

W


Example 7

Example 7

S is (5, -9) and T is (5, 12).

  • mST = (y2-y1) (x2-x1)

  • = (12 -(- 9))

  • ( 5 - 5)

  • = 21/0

  • and this is undefined

  • or infinite

NB: line looks like

T

Vertical line has infinite gradient.

S


Summary of gradients

Summary of Gradients

Positive gradient goes uphill.

Negative gradient goes downhill.

Zero gradient is horizontal.

Infinite gradient is vertical.


Parallel lines

Parallel Lines

  • Parallel lines run in the same direction so must be equally steep.

  • Hence parallel lines have equal gradients.

  • Example 8

  • Prove that if A is (4,-3) , B is (9,3) C is (11,1) & D is (2, -1)

  • then ACBD is a parallelogram


Ex8 ctd

Ex8 ctd

D

B

NB; The order of the letters is important.

A

  • mAC = (1 + 3)/(11- 4) = 4/7

  • mDB = (3 + 1)/(9 - 2) = 4/7

  • mAC = mDB so AC is parallel to DB

  • mAD = (-1 + 3)/(2 - 4) = 2/-2 = -1

  • mCB = (3 - 1)/(9 - 11) = 2/-2 = -1

  • mAD = mCB so AC is parallel to DB

  • Since the opposite sides are parallel then it follows that ACBD is a parallelogram.

C


The gradient formula

COLLINEARITY

Defn: Three or more points are said to be collinear if the gradients from any one point to all the others is always the same.

Example 8a

K is (5, -8), L is (-2, 6) and M is (9, -16). Prove that the three points are collinear.

6 - (-8)

14

= -2

mKL =

=

Since KL & KM have equal gradients and a common point K then it follows that K, L & M are collinear.

-2 - 5

-7

-16 - (-8)

-8

mKM =

=

= -2

9 - 5

4


The gradient formula

Ex8b

A Navy jet flies over two lighthouses with map coordinates (210,115) & (50,35). If it continues on the same path will it pass over a yacht at (10,15) ?

m1 = (115-35)/(210-50)

= 80/160

= 1/2

m2 = (115-15)/(210-10)

= 100/200

= 1/2

Since gradients equal & (210,115) a common point then the three places are collinear so plane must fly over all three.


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