It so happens that the triangle whose sides are 4, 5, and 6 has a special property: Its largest angle is twice its smallest angle. Make calculations that support this assertion.

## Set 05

## Cylinders, Pouring Contents

When the contents of a cylinder are poured into a second cylinder whose radius is 2 inches greater, the height reached in the second cylinder is one half of that reached in the first. Find to the nearest tenth of an inch the radii of the two cylinders.

## Trapezoid, Similar Triangles

Let *ABCD* be a trapezoid with the measure of base *AB* twice that of base *DC*, and let *E* be the point of intersection of the diagonals. If the measure of diagonal *AC* is 11, then what is the measure of segment *EC*?

- 11/3
- 15/4
- 4
- 7/2
- 3

## Line Splits Triangle Into Equal Areas

A vertical line divides the triangle with vertices (0,0), (1,1) and (9,1) in the *xy*-plane into two regions of equal area. What is the equation of the line?

*x*= 2.5*x*= 3.0*x*= 3.5*x*= 4.0*x*= 4.5

## Calculating With Supplements

In the diagrams below, the points *A* and *B* are located anywhere along the rays forming ∠*C*, whose measure is always 40°. The points *A* and *B* are connected by a line segment to form △*ABC*. The point *D* is the intersection of the bisectors of the exterior angles at *A* and *B*.

*ADB*?

## Sum of Slopes of Sides of Triangle

The base of an equilateral triangle is on the *x*-axis. What is the sum of the slopes of the three sides of the triangle?

## Find f(7) in 7th Power Equation

Let *f(x)* = *ax*^{7} + *bx*^{3} + *cx* – 5, where *a*, *b*, and *c* are constants.

If *f*(-7) = 7, then what is *f*(7)?

- -17
- -7
- 14
- 21
- not uniquely determined

## Defined Operations 1; Do Composition

Suppose the operations “#” and “*t*” are defined so that:

6 # = 20; 2 # = 4; 10 # = 36; and 5 *t* = 4.5; 10 *t* = 7; 8 *t* = 6.

Then what is *n* if 3 # # *t* = *n*?

## Birthday Present (Dimes, Quarters)

Hartley and Bertha are pooling their money to buy their mother a birthday present. (Her birthday, by coincidence, turned out to be the day she was elected to the school board.) The two have $20.25 between them. Hartley has all of his money in dimes and Bertha’s is in quarters. Hartley has twice as many dimes as Bertha has quarters. How much money is Hartley contributing?

## Columbus to Cincinnati

At 12:00 noon, Lunis started driving from Columbus to Cincinnati at a steady 40 mph. Fifteen minutes later, at 12:15, Ethan started driving from Cincinnati to Columbus at a steady 50 mph. The two cars passed each other at a point midway between the two cities. Using this information, calculate the distance between Columbus and Cincinnati. (Is this distance correct?)