O-Level Additional Math

[tweet]

Hi Guan Hui


Pls help with these 2 questions:

1) Given that A is the point (0,2), B is the point (9,0) and C is a point on the line y=2x + 2 such that AB = BC, find the coordinates of C. If ABCD forms a parallelogram, find the coordinates of point D.
Answers: C(2,6) and D(-7,8 )

2) Find the area of triangle ABC whose coordinates are A(3,3), B(-1,0) and C (5,-3) Hence, or otherwise, calculate the distance from C to AB.
Answer: 15;6

TIA

[Guan Hui]

zzz… typing this for the 3rd time… haha here we go


let c be (x,y)
distance of ab=sqrt(9^2+2^2)
=sqrt(85)
distance of bc=sqrt((9-x)^2+y^2)
since ab=bc
sqrt((9-x)^2+y^2)=sqrt(85)
(9-x)^2+y^2=85
y^2=85-81+18x-x^2
y^2=4+18x-x^2------------------1
y=2x+2
y^2=(2x+2)^2
y^2=4x^2+8x+4------------------2
4+18x-x^2=4x^2+8x+4
5x^2-10x=0
x=0 (rejected cause this value is for a)
or x=2
put x=2 into y=2x+2
y=6
c(2,6)

since abcd is parallelogram, ab ll cd and bc ll ad
gradient ab=(2-0)/(0-9)
=-2/9
y =-2/9x +c
sub c in,
6=-4/9+c
c=58/9
equation of cd: y=-2/9x+58/9

gradient of bc=(0-6)/(9-2)
=-6/7
equation of line ad: y= -6/7x +2

-2/9x+58/9=-6/7x+2
-14x+406=-54x+126
40x=-280
x=-7
put x=-7 into y=-6/7x+2
y=8

d(-7,8 )

[Guan Hui]

ab=sqrt((3+1)^2+3^2)

=sqrt(25)
bc=sqrt((-1-5)^2+(0+3)^2)
=sqrt(45)
ac=sqrt((3-5)^2+(3+3)^2)
=sqrt(40)
let the angle at point c be x
cosx=( ac^2+bc^2-ab^2 )/(2 acbc)
=60/2 sqrt(1800)
=sqrt(0.5)
x= 45 degree
sin 45 =sqrt(0.5)
area = 0.5 bc ac sin 45
=15

since ab= sqrt 25= 5
area of triangle:
0.5 x 5x h =15
h=6

**note
sqrt (1800)= sqrt (36000.5
= 60 sqrt (0.5)

1/2sqrt(0.5)=sqrt (0.5)

[tweet]

Hi Guan Hui


Thanks for your help.Will show the worked solution to my son.

Rgds

[emerald]

Hi, need your help to solve the following using Commutative Laws, Associative Laws &/or Distributive Laws (Easy method):


(54.2 x 1.8 + 0.2 x 25.86)


TIA.

[Guan Hui]

54.2x1.8+0.2x25.86

= 54.2x9x0.2+25.86x0.2
=0.2(54.2 x 9+25.86)
=0.2(50x9+4.2x9+25.86)
=0.2(450+37.8+25.86)
=0.2(513.66)
=102.732

I do not know if this is the answer you want emerald.
For me I will jus take 54.2 x 1.8 and 0.2 x 25.86 it will be much easier.

For multiplication of decimals just need to remember these 2 examples:
54.2 x 1.8(take not 54.2 have 1 decimal place and 1.8 have 1 decimal place also)
Total decimal place =1+1= 2
ignore the dots....
542 x 18= 9756
bring in the dot... 2 decimal place
ans=97.56

another example:
0.1 x 0.01 (total decimal 3)
remove dots... 1x1 =1
put in dot 3 decimal place..
ans: 0.001

hope it helps

[emerald]

Thanks a lot, Guan Hui.

[OK Lor]

Hi Guan Hui,


Please help to solve:

x²-286x+c=0 has roots a²b and ab². x²+px+q=0 has roots a and b. Find p and q, given that a, b and c are integer positive.

Thanks.

[Guan Hui]

if x²-286x+c=0 have root a²b and ab²

means
(x-a²b)(x-ab²)=0
expand it
x²-ab²x-a²bx+a^3b^3=0
x²-(ab²+a²b)x+a^3b^3=0
so…
ab²+a²b=286------1
a^3 b^3=c---------2

x²+px+q=0
(x-a)(x-b)=0
x²+x(-a-b)+ab=0
-a-b=p--------------3
ab=q----------------4



ab²+a²b=286
ab(a+b)=286
since a b and c are integer positive
ab and (a+b) are integer positive
factors of 286
1x286
2x143
11x26
13x22

13=(11+2) while 22 =(11x2)
a=11 or 2 and b=2 or 11
c= a^3b^3=11^3 2^3
c=10648

p=-a-b
p=-13
q=11x2
=22

sorry for late answer just saw it this morning.

[OK Lor]

Hi Guan Hui,


Thanks a lot, cheers.