Solutions to Quiz - 3-8-08

Problem 1

Problem 2.

Problem 3.

  A direct attack on this problem is by using the coordinate system.

Let's take BC as the x-axis, AB as the y-axis, point B as the origin. We just need to find the coordinate (x,y) of point O. This is rather straightforward.

x^2 + (y - 6)^2 = 50    ... (1)
y^2 + (x - 2)^2 = 50    ... (2)

Solving for (x, y) , we get the solution as:   (-5, 1) or (7, 1),  of which (-5,1) is the realistic solution.

Thus we have the solution for the length of BO  as  sqrt(26).

Problem 4.

Denote length of AB as |AB|, and let 
|AB| =  x  and  |BC| = y.

It's easy to see from the top graph

( x - 21) : 21 = 21 :  ( y - 21)    ... (1)

From the 2nd graph

|AE| = sqrt(440) * x / y
|FC| =  sqrt(440) * y / x

 

|AC| = |AE|+|EF|+|FC| gives
sqrt( x^2 + y^2 ) = sqrt(440) ( 1 + x/y + y/x)                                 ...  (2)

I'll leave you to solve the set of equations (1) and (2) for x and y.