Well, my teacher gave a solution , but as my trig is horrible , ididnt understand it.....
now we know $\displaystyle angles PAB , PBC, QBC, QCB, RAC, ACR = 30 $
Now $\displaystyle AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30 $ , which implies $\displaystyle AP+CQ = AR $ OR, AP+CQ-AR=0
It may be very trivial but i dont understand why $\displaystyle AP \sin 30 + CQ \sin 30 = \frac{1}{2} AC = AR \sin 30 $
please help!
Note that $\displaystyle \displaystyle \triangle APB$ and $\displaystyle \displaystyle \triangle BQC$ are both isosceles triangles with the same side lengths. Similar triangles with exactly the same magnitude. (do you see why?)
B bisects AC (why?), and so, the bases of triangles $\displaystyle \displaystyle APB$ and $\displaystyle \displaystyle BQC$ are each half the length of $\displaystyle \displaystyle \triangle ARC$, which is also a similar triangle. Since these are similar, corresponding sides are proportional, and knowing $\displaystyle \displaystyle \triangle ARC$ is double the magnitude of the two other triangles, we know the exact proportion. The upshot of all this is:
$\displaystyle \displaystyle AP + CQ - AR = \frac 12AR + \frac 12 AR - AR = 0$
since AP and CQ are corresponding sides to AR with half its magnitude.
Now I spouted a lot of unproven claims here. It's up to you to fill in the blanks
Fine, so we want to use trig. Very well. We know the magnitude of all the angles because we see that we had similar triangles here, they are all isosceles, and hence the other angles were 30 degrees each.
construct two perpendicular lines to AC, one passing through P, the other through Q. That will give you the sine relationships your professor mentioned (Remember SOHCAHTOA!) The fact that the sum is 1/2 AC goes back to what I was saying about similar triangles with the factor of 1/2
Hello jhevon,
A point on a line can be placed anywhere if there is no spec given.To draw a diagram for a proof lets make AB= 4 cm BC = 6 cm AC =10cm. Points P, Q, R lie on the perpendicular bisectors of AB,BC, AC respectively. Three 30-30-120 isosceles triangles are erected as described. Extend AP and QC to meet the perpendicular bisector of AC at S above AC
ARCS and PBQS are parallelograms AP+QC = AR.The whole is the sum of the parts
yes ...
sorry if i am being a moron ,but ....
suppose AP and QC when extended meet as S. Can we be sure that the perpendicular bisector of AC always passes through the point S ???Originally Posted by bjhopper
another doubt: is the original trig solution ok?
HELP!