Circle In An Isosceles Triangle

This type of problem has been showing up more and more often on the Calculator test. I have personally seen it on a region test, district test, and a few invitational tests.

If the circle is inside the isosceles triangle, then the radius can be found using 1/2*b*sqrt((2a-b)/(2a+b)) where a is the smallest side and b is one of the congruent sides.

-Kevin

Law of Cosines

This is typically used for problem number 40.

When given a triangle with two sides a and b and the included angle, the remaining side can be calculated using the law of cosines:   

Example:     

 

-Kevin

Scaling Principles

 Principle 1: LengthIf x and y are length dimensions on geometrically similar figures, then y is proportional to x, then
 

Principle 2: Area
If x is a length and A is an area on geometrically similar figures, then

Principle 3: Volume

If x is a length and V is a volume on geometrically similar figures then

Example:

xin/9.4in=273cm/100cm = 25.662 in shadow * 2.54*10 = 652 mm (2.54 cm in 1 inch)  

Kevin Valle :D

Percents

Percent Change/Difference = 100 * [(B/A) - 1] where B is the 2nd number given and A is the 1st.

Example:

 

B = area of letter-sized sheet = 8.5*11 = 93.5
A = area of legal-sized sheet = 8.5*13 = 110.5

% change = 100 * [(93.5/110.5)-1] = -15.4 or -1.54x10^1

Percent Error = 100 * [(approximate/exact) -1]

Example:

Approximate = 1152
Exact = 1176.37
Percent Error = 100 * [(1152/1176.37)-1] = -2.07 (notice that this is an SD problem and is 3 sig figs)

-Kevin

Geometry Problems

The geometry problems follow the same format on every Calculator test.
Problem Numbers 9, 10: Simple, usually one step, calculations involving circles, rectangles, triangles, trapezoids, squares, rhombus, etc.
Problem Numbers 19, 20: Right triangles featuring sin, cos, tan, and the pythagorean theorem.
Problem Numbers 29, 30: Simple calculations involving three-dimensional figures.(Frustrums, spheres, cones, pyramids)
Problem Numbers 39, 40: Inscribed and circumscribed circles, law of sines, law of cosines
Problem Numbers 49, 50: More complex three-dimensional figures, that may be combined to make one figure.
Problem Numbers 59, 60: Graphs using calculus and REALLY complex shapes.
Problem Numbers 69, 70: Straight from the Calculator Applications Study Guide available on the UIL

Unit Conversions

These are useful all throughout the Calculator test, along with the Math and Number Sense tests as well. It is best to have most of these conversions memorized. Although they are not asked for directly, these equations are necessary to complete many types of problems.

1 inch = 2.54 cm.
1 mile = 5280 ft.
1 sq. mile = 640 acres
1 acre = 43560 sq. ft.
1 acre = 4840 sq. yds.

1 teaspoon (tsp) = 1/6 oz.
1 tablespoon (tbsp) = 1/2 oz.
1 cup = 8 oz.
1 pint = 16 oz.
1 quart = 32 oz.
1 half gallon = 64 oz.
1 gallon = 128 oz.
1 cubic foot = 7.481 gallons (approx.)
1 liter = 1.0567 quarts (approx.)
1 pound = 16 oz.
1 pound = 453.59237 grams
1 ounce = 28.3495231 grams

Radius of the Earth = 3960 miles

Degrees Celsius = (Degrees Fahrenheit - 32 degrees)/9 = Kelvin - 273.15

Scalene Triangles

r = (s-c)*tan(C/2) = [(a+b-c)*tan(C/2)]/2
R = (abc)/4K
K = sqrt(s*(s-a)*(s-b)*(s-c))