and are noncollinear rays (endpoint A). Rhombi are drawn, having one vertex at A, one vertex on , and one vertex on . What is the locus of the fourth vertex?

## Set 07

## Triangle Rotating in Square

Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:

- 20/3
- 32/3
- 12
- 40/3
- 15

## Conjecture About Angles

Acute angle *B* has *n* degrees. Consider this statement: “If angle *B* is subtracted from its supplement, then the result is twice the complement of angle *B*.”

- Give
*n*several values, then see whether you think the sentence is true. - Show algebraically that the statement is true or not true.

## Collinear Vertices of Squares

If the three vertices A, B, and C in the three adjacent squares shown are collinear, find the value of *x*.

## Cubic to Quadratic to Linear

One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form *ax*^{3} + *bx*^{2} + *cx* = *d*

## Find f(x+1) – f(x)

If *f*(*x*) = 4^{x}, then *f*(*x* + 1) – *f*(*x*) =

- 4
*f*(*x*)- 2
*f*(*x*) - 3
*f*(*x*) - 4
*f*(*x*)

## Brick Wall Without Picture

Tim has employed Kathy and Leslie as bricklayers for the summer. Chris wants a brick wall built between his diving pool and the main swimming pool. Tim estimates that Leslie could build the wall in 9 hours and Kathy could build it in 10. However, he has learned that when they work together their combined output decreases by 10 bricks per hour. But, being in a hurry, he puts them both to work on it and finds that it takes them exactly 5 hours, working together, to finish the wall. How many bricks are in the wall?

## Driving to Cleveland

Zach and his family were driving into Cleveland for the ball game. Zach fell asleep when they were halfway along the way. When he woke up the distance they still had to go was half as far as they went while he was asleep. For what fraction of the way did Zach catch his Z’s?

## Intersecting Lines

*m*are the simultaneous equations

*y*=

*mx*+ 3

*y*= (2

*m*– 1)

*x*+ 4

*x*,

*y*)?

- all
*m*> 0 - all
*m*≠ 0 - all
*m*≠ .5 - all
*m*≠ 1 - all values of
*m*

## Cat Burglar, Crate

It is night. You are in a quiet city, near the sea. There is no moon. Near a wharf, a cubical wooden shipping crate is lying along the outside wall of a warehouse, flat against the wall. Each edge of the crate is 1 meter. At midnight, out of the crate emerges Cathy, the notorious cat burglar. She noiselessly leans a √15 meter ladder against the wall in such a way that it just grazes the free horizontal edge of the crate. At what height does the ladder hit the wall?