I've been trying to solve this problem for a while now, but I still cannot do it. I understand that AB has a length of 5cm, and so if BC is "x" then AC = 5 + x. From there, I'm stuck. Please can somebody show me how to do it. Thanks!
I've been trying to solve this problem for a while now, but I still cannot do it. I understand that AB has a length of 5cm, and so if BC is "x" then AC = 5 + x. From there, I'm stuck. Please can somebody show me how to do it. Thanks!
well first you need to find the angle at which the line AB makes with the horizontal (angle of ABX)
from the picture i attach, angle of ABX can be found by:
tan ABX = (AE - BD) / ED
tan ABX = (radius of larger circle - radius of smaller circle)/(radius of larger circle + radius of smaller circle)
tan ABX = (4-1)/4+1)
tan ABX = 3/5 = 0.6
ABX = 30.9637 degrees
from observation this will also be the angle of ACE:
sin ACE = AE / AC
sin 30.9637 = (radius of larger circle) / AC
AC = radius of larger circle/sin 30.9637
AC = 7.774603526 cm
ahhh i see my mistake now sorrie >.<
it is suppose to be
sin ABX = (radius of larger circle - radius of smaller circle)/(radius of larger circle +radius of smaller circle)
sin ABX = 4-1/4+1 = 0.6
again using observation angle ACE = angle ABX
sin ACE = radius of larger circle/AC
0.6 = 4/AC
AC = 4/0.6
lol thats where i made my mistake. i mistakenly assumed that AB = ED however it is not actually equal because the circle is not drawn up to scale. the smaller circle is actually very much smaller.
from the attachment AY is not equal in length to EZ. The only possibility for AB = ED is when both circles are equal in size(same radius)
well they are using the same method but more vaguely.
consider the triangle BCD and triangle ACE. obviously we can see that triangle BCD is the smaller version of triangle ACE.
from the attachment,
for the smaller triangle,
cos theta = BD/BC ====(1)
for the larger triangle,
cos theta = AE/(AB+BC) ======(2)
from (1) and (2)
BD/BC = AE/(AB+BC)
BC/BD = (AB+BC)/AE
let BC = x
x/1 = ((4+1)+x)/4
4x = 5+x
3x = 5
x=5/3 = BC
now AC = AB + BC = 4+1+5/3
=5+ 3/3 + 2/3
=6 + 2/3
well you already got the thinking right. the triangle ABX has a hypotenuse of AB and hypotenuse are always longer than any other side length of the triangle.
For exampleattachment)
sin theta = 8/10 = 0.8 ===> means that the line OB is a fraction of line AB
sin alpha = 6/10 = 0.6 =====> means that line OA is a fraction of line AB
this is true for all triangles
in our case A=A, B=B,O=X.
cos ABX = BX/AB ====> line BX is a fraction of length of line AB(BX<AB)