The

Power of a Point PowerPoint

(with some other kinds of problems too)

April 1, 2009

Quadrilateral ABCD is circumscribed about circle O. Find the perimeter of ABCD.

Answer to Quadrilateral ABCD:28

Solve the equation log2x216 =x, where x is real.

Answer to log2x216: x = 3

If the sum of two complex numbers is 1 and their product

is 1, what is the sum of their squares?

Answer to sum of squares:-1

Hexagon ABCDEF is circumscribed about a circle.

AB = 8, CD = 9, EF = 10, and BC = 7.

Find the value of DE + FA.

Answer to Hexagon ABCDEF:20

Find the shaded area.

Answer to shaded area:30

If 19C8 (“19 choose 8”) is 75,582,

what is 19C9?

Answer to 19C9: 92,378

Find the center of the circle with equation

x2 + y2 - 6x + 4√2 y = 64.

Answer to the center of the circle: (3, -2√2)

What base 8 number does 1100102 represent?

Answer to base 8 number:628

AC = 4, CD = 5, radius OD = 18. Find AB.

Answer to AB: 171

4

Find the product of the roots of 6x2 + 17x – 42 = 0

Answer to product of the roots:-7

Find the constant term of the expansion of (x2 – 2/x)6

Answer to the constant term: 240

A circle of radius 2 rolls around the outside of a square of side 4. Find the length of the path

made by the center of the circle.

Answer to the circle rolling around the square:

16 + 4π

In how many distinct ways can the letters in LJUBLJANA be

arranged?

Bonus for double points: Of what country is Ljubljana the

capital?

Answer to the number of ways 45,360

Bonus answer: Slovenia

The End of Ciphering

Find the fallacy in each of the following April Fools problems.

Three traveling salesmen stop at an inn. There is only one small room left. They are tired and take it anyway. The room is $30, so each of the 3 contributes $10. In the morning the manager arrives and decides to give them a partial refund. He gives the bellboy $5 to give to the salesmen. The bellboy realizes the men don’t expect a refund, so he gives them back only $3 and keeps $2 for himself. The men split the refund, taking $1 each. As each man had originally paid $10, but received $1 back, it ended up costing each man $9. They are happy with this and the bellboy is happy as he has $2 in his pocket.Question: each of the 3 men ended up paying $9. 3X9=27+2 (money in bellboy’s pocket) = 29. We started with $30. What happened to the extra $1?

Step 1: Let a=b. Step 2: Then a2 = ab, Step 3: a2 + a2 = a2 + ab , Step 4: 2a2 = a2 + ab , Step 5: 2a2 - 2ab = a2 + ab -2ab, Step 6: 2a2 - 2ab = a2 - ab Step 7: This can be written as 2(a2 - ab) = 1(a2 – ab) Step 8: and canceling the (a2 – ab) from both sides gives 1=2.

1: -1/1 = 1/-1 2: Taking the square root of both sides: √(-1/1) = √(1/-1) 3: Simplifying: √(-1) / √(1) = √(1) / √(-1) 4: In other words, i/1 = 1/i. 5: Therefore, i / 2 = 1 / (2i), 6: i/2 + 3/(2i) = 1/(2i) + 3/(2i), 7: i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) ), 8: (i2)/2 + (3i)/2i = i/(2i) + (3i)/(2i), 9: (-1)/2 + 3/2 = 1/2 + 3/2, 10: and this shows that 1=2.

Step 3 is wrong. The problem is that there is no rule that guarantees √(a/b) = √(a) / √(b), except in the case in which a and b are both positive.

If this surprises you, think about the questionWhy should √(a/b) equal √(a)/√(b) ?

If you were to try to convince someone of this, you'd have to start with the

definition of what a "square root" is: it's a number whose square is the number you started with. So all that has to be

true is that √(a) squared is a, √(b) squared is b, and √(a/b) squared is a/b.

So, when you square √(a/b), you will get a/b, and when you square √(a)/√(b), you will also get a/b. That's all that the

definition of square root tells you.

Now, the only way two numbers x and y can have the same square is if x = ±y.

So, what is true is that √(a/b) = ± √(a)/√(b), but in general

there's no reason it has to be √(a/b) = +√(a)/√(b),

rather than √(a/b) = -√(a)/√(b), unless a and b are both positive, for

then (because by convention we take the positive square root) everything in

the above equation is positive.

In our case, it is true that √(-1/1) = √(-1) / √(1), but √(-1/1) (that is, i)

is -√(1)/√(-1) (that is, -1/i) not +√(1)/√(-1) (that is, 1/i)

The fallacy comes from using the latter instead of the former.

In fact, the whole proof really boils down to the fact that

(-1)(-1) = 1, so √ (-1 * -1) = 1,

but √(-1) * √(-1) = i2 = -1 (not 1). The proof tried to claim that these two were equal (but in a more disguised way where it was harder to spot the

mistake).