# ACT, Geometry – diagonal of a rectangular prism

Q: What is the (straight-line) distance between points A and B on the figure below? Explanation:  We are finding the length of the green segment below: This requires the three-dimensional version of the distance formula:

There is another way to do this, using a progression of two right triangles, with the green segment being the hypotenuse of the second triangle, but I find the above formula much simpler.  It is just an extension of the ‘usual’ (2-dimensional) distance formula we are so familiar with from Algebra and Geometry.

BUT if you want an even easier way, and I know you do, use my TI-84 Plus program ‘DIST3D’. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Geometry – polygons

Q: A hexagon has interior angles measuring (in degrees) 137, 100, 150, 114, 90, and x. What is the value of x?

Explanation:
The total measure of all the interior angles in a polygon is (n-2) * 180, where n is the number of sides of the polygon.
A hexagon has 6 sides, so n = 6. The total is (6-2) * 180 = 720.
Add up the five given angle measures: 137 + 100 + 150 + 114 + 90 = 591.
The measure of the remaining angle is 720 – 591 = 129.
My TI-84 Plus program POLYGON will find this total for you. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Algebra 1 – Find the equation of a line

Q: If the slope of the line through (-3,6) and (1,y) is 1/2, what is the value of y?

Explanation: Slope is computed by the formula (y2 – y1) / (x2 – x1) … the change in y divided by the change in x.

So,

(y – 6) / (1 – -3) = 1 / 2

(y – 6) / 4 = 1 / 2

Multiply both sides by 4 and you have:

y – 6 = 2

y = 8

BUT, if you’re taking a test and having trouble remembering the method above, try this instead: Plot the point (-3,6) on a graph and move from that point with a slope of 1/2. Move up 1, right 2, up1, right 2… and repeat this pattern until your x-value is 1. Then, read the y-value at the same point. This is not my favorite way, but it works!

And one more thing!  If this were a multiple choice question, you could use my TI-84 Plus program ‘DISTANCE’ (which finds distance, midpoint, and slope for any two ordered pairs).  Substitute one of the answer choices for x, and see if the program reports that slope is 1/2. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Algebra 1, 2 – systems of equations

Q: Red pens cost twice as much as blue pens. Two blue pens and three red pens cost a total of \$2.32. How much does a blue pen cost?

Explanation:  Let r be the cost of a red pen, and let b be the cost of a blue pen.

Then, r = 2b and 2b + 3r = 2.32. Solve this system of equations.

By substitution,

r + 3r = 2.32

4r = 2.32

r = 0.58

Plug r= 0.58 into the first equation (or second, but I like the first better):

0.58 = 2b

0.29 = b

You can also use other algebraic methods such as elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program CRAMER once you rearrange the system to say: 2b – r = 0 and 2b + 3r = 2.32. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – Counting and Ordering

Q:  20 students are to choose a president and vice president from among themselves. How many possible ways can they be chosen?

Explanation 1:  Using the Counting Principle, set up two blanks to represent the president and vice president, from left to right:

____    ____

There are 20 choices for the president.  Put 20 in the first blank:

_20_    ____

The same person cannot be chosen for both positions, so there are 19 left to choose from for the vice president.  Put 19 in the second blank:

_20_    _19_

Now, multiply these numbers.  20 x 19 = 380.

Explanation 2:  This is a permutation problem since you are choosing 2 people from a total of 20 people, and the order is important.  The order is important because the two people chosen must be given specific positions, and switching those positions results in a different scenario.  Do  20 nPr 2 , or use the formula for nPr.  On the TI-84 Plus, find nPr in MATH-PRB-2.  The result is 380. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT, Algebra 1 – percent change

Q: Last year, Emmy was 40 inches tall.  This year, she is 42 inches tall.  By what percent has her height increased?

Explanation:  Percent change is calculated by the following formula:

(New – Original)/Original  x 100%

(42 – 40)/40 = 2/40 = .05

.05 x 100% = 5%

Since it is positive, this is a percent increase.

My TI-84 Plus program PRCNTCHG will calculate this, too. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT – number properties

Q: What is the sum of all of the integers from 1 to 100?

Explanation: Don’t add 1 + 2 + 3 + 4 + … all the way up to 100! While it will work, it is a giant waste of time. Look for a pattern of numbers to pair up:

1 + 99 = 100

2 + 98 = 100

3 + 97 = 100

49 + 51 = 100

That’s 49 one hundreds. 49×100 = 4900. Now, add the numbers we have not counted yet: 100 and 50.

4900 + 100 + 50 = 5050.

My TI-84 Plus program ‘SUMATOB’ will find this sum, too. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# Algebra 2 – Sequences

Q: What is the tenth term of the sequence: 4, 12, 36, 108, …?

Explanation:  It’s a geometric sequence because each term is multiplied by 3 to get the next term. The common ratio (r) is 3. You *could* just keep multiplying by 3 until you get to the tenth term. It’s not the most efficient way, but it works.

The ‘best’ way, if you know it, is to use the formula for the nth term of a geometric sequence: Oh, and my TI-84 Plus program ‘NTHTERM’ will find this answer for you. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

# SAT / Algebra 1: Ratios

Q:  In my box of cereal, the ratio of flakes to raisins is 9:2. If there are a total of 220 flakes and raisins in the box, how many of these are flakes?

Explanation:  Solve the equation: 9x + 2x = 220 to get x = 20.  ‘Flakes’ are the 9x part of the equation.  9(20) = 180.  Or, use my TI-84 Plus program ‘RATIO’.  