FURTHER MATHEMATICS SS 1 SECOND TERM
CLASS: SSS 1
SUBJECT: Additional Maths
TOPIC: THE STRAIGHT LINE
CONTENT: (I) mid- level of a section (ii) Gradient of a straight line (iii) Distances between two factors (iv) Situations for parallelism and perpendicularity (v) angles between two intersecting traces.
SUB-TOPIC: MID-POINT OF A LINE SEGMENT
Within the Cartesian aircraft above, let be the mid-point of the road section with the coordinate and .
As triangles are related;
Since R is the mid-point,
Therefore, the co-ordinates of the mid-point of the road becoming a member of and are:
Instance 1:- Discover the mid-point ‘R’ of the road section the place and .
The mid-point of a line section is a particular into ratio; On this case, the ratio is //
The next are different circumstances of dividing a line section in given ratios :
- Inner division:
Let and be the 2 given factors on a line section which divides it within the given ratio . It’s required to search out the co-ordinates of P. Suppose they’re , as illustrated within the cartesian graph sketch beneath:
With the letterings on the graph and related triangles
Now, AP: PB=M1:M2,
AK = MN = ON – OM = X2-X
KP = MP- MK = MP- LA = y-y1
TB = NB – NT = NB – MP = y2-y
From (i) we now have
The primary two relation give
Equally, from the relation we get
this finally offers
Therefore, the co-ordinate of the purpose diving a line becoming a member of () and within the are given as ,
Be aware that these outcomes applies to increase division, with both or taken as destructive
Discover the co-ordinates of the purpose which divides the road becoming a member of the factors (8,9) and (-7,4) internally within the ratio 2:3 .
The co-ordinates of the purpose is obtained by substitution,
Instance 3: Discover the centroid of the triangle whose vertices have the coordinate (-4,6),(2,-2) and (2,5) respectively.
Answer:- Recall :
The centroid of a triangle is the purpose of intersection of its median.
Let AD be the median bisecting its base.
The purpose on which divides it internally within the ratio 2:1 is the centroid. If are co-ordinates of the centroid, then
The centroid is (0,3)
One other case of inner division is as proven within the section beneath
We are saying A divides PQ internally within the ratio 3:2
- Everlasting division
We are saying A divides PQ externally (i.e exterior PQ) within the ratio 3 : -1
i.e (1<3, subsequently A lies on the left of P)
divides externally within the ratio -1:3
The purpose C divides the road the place the co-ordinates of respectivelly within the ratio 3:2 .Discover the co-ordinate
Therefore, the co-ordinates of C are (6,1)
- Discover the co-ordinates of the mid-points of the traces becoming a member of the next pairs of factors;
- (3,6) and (5,8) (b) () and ()
- Discover the mid-points of the perimeters of triangle whose vertices are
Sub-Subject 2: Gradient of a straight line
The gradient of a line is outlined because the ratio enhance in in going from one level to a different on a line.
Is the change in x because the variable x will increase or decreases from x1 to x2 and is the change in y with respect to y1 and y2 .
The slope (gradient) m of a straight line L is outlined as
If is the angle of inclination to the slope of L,then ; is known as the angle of slope of the road.
Instance 5: Discover the slope and the angle of inclination of the L by means of factors and
The slope m of factors P1 and P2 on L is
The slope of the factors P1 and P3 on L is
Subsequently, implies that the slope of the road L is 3.
Subsequently, the angle of inclination is 71.57
It may possibly subsequently be concluded from the instance above that any given line has one and just one slope.
Be aware: the factors and are mentioned to be collinear that means that the factors lie on a straight line, and the gradient of any two of the purpose is identical.
Instance 6: Discover the gradient of the road becoming a member of the pair of level
Let symbolize the gradient
- Discover the angle between traces L1 , with slope -7 and L2 which passes by means of (2,-1) and (5,3)
- Discover the gradients of the traces becoming a member of the next pairs of factors :
Sub-Subject 3:- DISTANCE BETWEEN TWO POINTS
Let be two distinct factors. The gap between them can thus be calculated.
Making use of Pythagoras theorem to right- angled triangle within the graph above,
Calculated the space between the factors, (4,1) and (3,-2)
The gap between the factors (4,1) and (3, -2) is 22
Discover the distances between the given factors:
SUB – TOPIC 4: CONDITIONS FOR PARALLELISM AND PERPENDICULARITY/ ANGLES BETWEEN TWO INTERSECTING LINEs.
Within the diagram above, the angles between the traces AB and DC are as indicated are associated as follows:
AQC = DQB = (vertically reverse angles)
21 in triangle ACQ (exterior angle of a triangle equals sum of reverse inside angles)
The traces AB (gradient m1) and CD(gradient m2) making angles 1 and 2 with the reveals that: 1 = m1 and 2 = m2
Since 2 = 121
⇒ 21) 2 – 1)21) =
i.e. if is the angle between two intersecting traces whose gradients are then the angle between them is
If the 2 traces are parallel, the angle between them is zero, therefore and therefore m2 – m1)/(1 + m2m1) ⇒m2 – m1) = 0 ⇒ m2 = m1
If the 2 traces are perpendicular, . Since any worth divided by zero⇒from m2 – m1)/(1 + m2m1), (1 + m2m1) = 0, m2m1 = – 1 or m1 = -1/ m2 or m2 = -1/m1
Instance 8: Discover the acute angle between the pair of traces and
Answer: let the gradients of the traces be m1 and m2 respectively.
Recall that ; subsequently, writing every of the traces on this kind we acquire
Therefore, m1 and m2
Subsequently, the angle between the 2 traces is given by
Instance 9: Discover the gradient of a line parallel to the road which passes by means of the pair of factors (-1,3) and (4,7)
Answer: let the gradient of the road passing by means of the factors (-1,3) and (4,7) be m1
Therefore, the gradient, m2, of a line parallel to the road which passes by means of the pair of factors (-1,3) and (4,7) is since m2 = m1
- Discover the gradient of a line parallel to the road whose gradient is 2.3
- Discover the gradient of a line perpendicular to the road which passes by means of every pair of the next factors
- (0,8) and (-5,2) (b). (-k,h) and (b, -f)
- Discover the inside angles of the triangle whose vertices are A(4,3), B(-2,2) and C(2,-8).
- The road becoming a member of (-5,7) and (0,-2) is perpendicular to the road becoming a member of (1,-3) and (4,x). Discover x.
- The gradient of the road passing by means of the factors Discover the worth of
- The gap between is 13units. Discover the values of
- Utilizing gradients, decide which of the next units of three factors are collinear?
- (1,-1) , (-2,4), (0,1)
- (5,-2) , (7,6), (0,-2)
- (-2,3) , (8,-5), (5,4)
- (6,-1) , (5,0), (2,3)
- (-1,5) , (3,1), (5,7)
- Discover the gradient of a line parallel to the road whose gradient is – 3
- Given the three factors
- The gradient of AB
- The equation of the road by means of C perpendicular to AB
- If M and N are the mid – factors of BC and AC respectively, calculate:
- The coordinates of the purpose of intersection G
- The worth of the ratio lAGl : lGMl.
- In parallelogram ABCD, slope of AB = -2, slope of BC = . State the slope of
- AD c. CD
- The altitude to AD d. the altitude to CD