Links to practice problems for distance between two points and area of a triangle defined by three points, one of which is (0, 0)

Here is the link to the practice problems.

More practice
Here are four points. Distances should be given in simplified square root form and approximated to the nearest thousandth,

(0, 0), (6, 3), (10, 1), (-4, -5)

a) Distance from (0, 0) to (6, 3)
b) Distance from (0, 0) to (10, 1)
c) Distance from (0, 0) to (-4, 5)
d) Distance from (6, 3) to (10, 1)
e) Distance from (6, 3) to (-4, -5)
f) Distance from (10, 1) to (-4, -5)

g) the area of the triangle defined by the points (0, 0), (6, 3) and (10, 1)
h) the area of the triangle defined by the points (6, 3), (10, 1) and
(-4, -5)

answers in the comments.
 

Simultaneous equation practice with Kramer's rule

6x + 5y = 18
2x - y = -4

3x + 7y = 12
2x + 5y = -18

Answers in the comments.

100 coin (or ticket) problems

a) 100 coins, all quarters and pennies, total = $13.72

b) 100 tickets, all children ($6) and adult ($12), total =$912

c) 100 coins, all quarters and nickels, total = $8.20

d) 100 coins, all dimes and pennies, total = $3.79

e) 100 coins, all dimes and nickels, total = $7.85

f) 100 coins, all quarters and dimes, total = $14.65

Answers in the comments

The Richter scale: From two readings, the relative strength and vice versa

In class last week, we learned how to find out how much stronger one quake is compared to another given the two Richter scale readings. For example, on June 28, the strongest quake in the U.S. was a 3.4 in Oklahoma, while the strongest in North America was a 5.4 in Niltepec, Mexico. What is the difference in levels of energy? Here are the steps.

Step 1: Subtract little from big. In our case, 5.4 - 3.4 = 2.0.
Step 2: multiply difference by 1.5. 2.0 * 1.5 = 3.0.
Step 3: Raise 10 to the power of the answer from Step 2: 10^3.0 = 1,000. The Mexican quake was 1,000 times stronger than the Oklahoma quake.

Let's ask the question in the opposite direction. Let's say we have a reading for a quake and we know another quake was x times stronger. Again, it will be a three step process, but now we will take the inverse of our three steps above in reverse order. Let's say we have a quake 350 times stronger than the one in Niltepec.  Here are our steps.

Step 1: Take the log of the strength multiplier. Log is the inverse of raising 10 to a power, just like addition is the inverse of subtraction and division is the inverse of multiplication. log(350) = 2.544...,
Step 2: divide the answer from Step 1 by 1.5 and round this answer to the nearest tenth. 2.544/1.5 = 1.696..., which rounds to 1.7. We round to the nearest tenth because the Richter scale rounds to the nearest tenth.
Step 3: Add the answer from Step 2 to the Richter reading we know. In this case, it would be 5.4+1.7 = 7.1, the reading of the stronger quake. If instead we were told a quake was 350 times weaker than Niltepec, it would be 5.4-1.7 = 3.7

Here are some practice questions. The answers are in the comments.

1. 16 times stronger than a 6.1
2. 250 times stronger than a 6.7
3. 8 times weaker than a 5.8

Notes and practice for Venn diagrams and conditional probability

This link is to a post on my statistics blog about what we did today in the lab handout. Like posts here on the Math for Liberal Arts blog, there are problems at the end with answers in the comments.

Link to practice with set theory and cardinality.

Here's a link to some set theory problems.

Order of operations practice

Using the rules of order of operations, find the values of the following expressions.

a) 2(3² - 5)

b) 2(3 - 5)² 

c) 2²(3 - 5)

d) 2 × 3² - 5 

e) (2 × 3)² - 5

f) 2(3² - 5²)

Answers in the comments.

Square roots

Squares and square roots are what we call inverse operations.  If we take a number x and square it, the value is x². If we take the square root of x², we get the value x. A square, of course, is a geometric shape with four equal sides and four 90° angles. If we know the length of the side and call it s, the area is s², measured in square units. If instead we know the area is a number we call A square units, the side length for all the sides is the square root of A, which I will write on the blog as sqrt(A).

I will repeat this many times in class. When we are doing these kinds of problems, squares give us answers that are measured in square units which represent surface area, but square roots produce answers that are measure in regular units, which is to say lengths, one dimensional measure.

The Pythagorean Theorem is one of the central ideas in math, and it deals with the sides of a right triangle, which means a triangle that has one right angle. (A right angle measures 90°.) It is standard to name the three sides a, b and c, where c is the long side, always opposite the right angle. We call the long side the hypotenuse and the short sides are the legs.  The formula connected to the theorem is

a² + b² = c²

We can re-arrange the letters in the two following ways

a²  = c² - b²

or

b² = c² - a²

What this means is that though there are three numbers in the formula, if you have the values of any two of them, you can find the value of the third. In math, we say this set of numbers has two degrees of freedom.  ("Degrees of freedom" is not the same as "degrees", the units we use to measure angles.) You are "free" to use any two numbers for the lengths a and b, but once those are chosen, we can find c by using the formula and the square root operation, which means c is determined by a and b and can't be just any old number.  Likewise, you are "free" to pick and two numbers and call them b and c (or a and c), provided that c is bigger than the other number.  But once you have chosen two numbers out of the three, the third number is forced to be a specific value by using the formula.

A number is a perfect square or a square number if it is the square of a whole number.  Here are the perfect squares less than or equal to 100.

0² = 0  The square root of 0 is 0.
1² = 1  The square root of 1 is 1.
2² = 4  The square root of 4 is 2.
3² = 9  The square root of 9 is 3.
4² = 16  The square root of 16 is 4.
5² = 25  The square root of 25 is 5.
6² = 36  The square root of 36 is 6.
7² = 49  The square root of 49 is 7.
8² = 64  The square root of 64 is 8.
9² = 81  The square root of 81 is 9.
10² = 100  The square root of 100 is 10.


If a number is not a perfect square, its square root will not be a whole number. In fact, it will be an irrational number, a number that cannot be written as a fraction p/q. This means that as a decimal representation, we never get a pattern of digits that repeats forever. For example, the square root of 2 is represented on your calculator as 1.414213562... Any representation of a square root that has a finite number of digits is only an approximation.  It is common to round square roots to three places after the decimal, which is to say the nearest thousandth. When I write an approximation on the blog, I will use the symbols "~=".


Example: sqrt(2) ~= 1.414


The square root of 2 is the number whose square is exactly 2. but 1.414 is not the exact value.


1.414² = 1.999396

Besides approximating square roots, which we will be doing with calculators, there is also the idea of simplifying square roots, which will involve factoring a number down to its prime factorization.

Here are some practice problems for simplifying square roots, a method we learned today in class.

Here are practice problems for using the Pythagorean Theorem to find the length of the third side of a right triangle if you already know the lengths of the other two sides.

Here are some more practice problems using the Pythagorean Theorem.

Here are two sides of a right triangle.  In these problems, c is always the hypotenuse and a and b are the short sides, also called the legs.  Find the missing side using the Pythagorean Theorem, write it as a square root in simplified form and give the approximation to the nearest thousandth.

1)  a = 7, b = 6, c = _________

2) b = 6, c = 7, a = __________


3)  a = 8, b = 6, c = _________

4) a = 6, c = 8, b = __________


The answers are in the comments.

Changing a repeating decimal to a fraction in lowest terms.

Using Blogger editing, it's a little tricky to put a bar over a group of numbers. Instead I am going to put brackets [] around the repeating part. To write .16161616...., I'll type .[16] and to type .1666666..., I'll type .1[6].

Sorry if it's a little hard to read.

Method for changing from repeating decimal to fraction in lowest terms.

1. Call the repeating decimal x.
2. Multiply x by a power of ten that has as many zeros as there are digits in the repeating part.
3. Subtract x from the bigger number, which will cancel out the repeating part.
4*. IF the subtraction gives you a decimal number, multiply by some power of ten so you get (whole number times) x = (some other whole number)
5. Divide both sides of the equation by the number multiplying x.
6. Reduce the fraction to lowest terms.

Examples.
Example #1: .[16] = 0.1616161616....
1. x = 0.16161616...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 16.16161616....
3. 100x - x = 16.1616161616... - 0.1616161616...
which reduces to 99x = 16.
step 4 isn't needed.
5. x = 16/99, which is reduced to lowest terms.

Example #2: .1[6] = 0.166666....
1. x = 0.16666...
2. Because the repeating part has one digit, multiply x by 10 to get 10x = 1.6666....
3. 10x - x = 1.66666... - 0.166666...
which reduces to 9x = 1.5
4. 1.5 isn't a whole number, so multiply by 10 on both sides to get 90x = 15.
5. x = 15/90, which is not in lowest terms.
6. 15/90 = 5/30 = 1/6.

Practice problems.

a) Find the fraction for .[35]
b) Find the fraction for .3[5]
c) Find the fraction for .3[54]

Answers in the comments.

Click on this link for more practice problems with solutions.

Percent and other scales based on powers of 10

Scales based on powers of 10: The most famous scale base on powers of ten in percentage, which really means "per 100". It is much more common to see "53% of the people agree with the president's plan" than ".53 of the people..." or "53 out of every 100 people...". Technically, all those phrases are saying the same thing, but percentage is the most popular.

To get a number based on a power of 10 scale, you take the small number, divide it by the big number and multiply by the power of ten, so it is  small/big*scale. Sometimes we need greater precision because the proportions are so small, the small number is tiny in comparison to the big.

When I ask a class what is the legal limit for blood alcohol while driving, invariably someone will say "point oh eight" and most people will agree. But .08 is wrong; .08 = 8%, and the correct answer is .08% = .0008. I don't blame the students. The number is badly represented and it is an easy mistake to make. Let's take a look at the number on other scales of 10.

.08 out of 100 is the same as
.8 out of 1,000 or
8 out of 10,000 or
80 out of 100,000

80 parts out of 100,000 is a tiny proportion. To give an idea, ounce of pure alcohol mixed into ten gallons of blood would give you 78 parts out of 100,000, and most people have between a half gallon and a gallon and a half of blood in their body, between 4 and 12 pints. The amount of alcohol in a person's blood stream that is over the legal limit is about the same amount of alcohol as found in a capful of mouthwash used after brushing your teeth.

We will look at the per 100,000 scale for another type of statistic, measurements of mortality rates.

Here are the number of homicides in some local cities in 2007.

Oakland: 124 homicides
Richmond: 28 homicides
San Francisco: 98 homicides

Clearly, comparing these numbers is misleading, because we know these cities have very different numbers of citizens, so the standard way to measure these statistics is the per 100,000 population scale, which we find by the formula

small/big x scale

which in this case is

(# of homicides)/(city population) x 100,000

Oakland's population in 2007 is estimated at 415,000, Richmond at 106,000 and San Francisco at 825,000, so the murder rates on this standard scale are as follows

Oakland: 124/415000 * 100000 = 29.9
Richmond: 28/106000 * 100000 = 26.4
San Francisco: 98/825000 * 100000 = 11.9

So even though more people were murdered in San Francisco than in Richmond in 2007, the murder rate in Richmond was over twice as high, because Richmond has barely 1/8 of the population of San Francisco. (note: The trends for the three cities this decade are going in different directions. Oakland's murder rate is on the rise, while Richmond's is falling and San Francisco's has stayed about the same.)

Practice problems: (answers given in comments)
1) Here are the homicide numbers for Oakland, Richmond and San Francisco from 2004.

Oakland: 96 homicides, 399,000 population
Richmond: 40 homicides, 99,000 population
San Francisco: 96 homicides, 775,000 population

Find the murder rates from these years, rounded to the nearest tenth per 100,000 population and rank them from lowest (1st) to highest (3rd).

Practice problems for first homework of Summer 2012

Change these numbers in Hindu-Arabic to Roman numerals.

648 = ______

15,781 = _____

Change these numbers in Roman numerals to Hindu-Arabic.
__
IXCDXLVII = ____

DCXCIV = ____

Find the approximation of 7/37 to the following number of decimal places and find the exact repeating decimal representation.

7/37 to the nearest tenth = _____
7/37 to the nearest hundredth = _____
7/37 to the nearest thousandth = ____

Repeating decimal representation = _____

Find the prime factorization of 48.

Find all the factors of 48.

Add or subtract these times, given in minutes and seconds.  If the minutes are 60 or more, change the answer to hours:minutes:seconds.

27:25 + 18:58 + 21:44 = __________

17:21 - 9:46 = __________

Answers in the comments.

Practice with square roots for homework due July 6.

The editor for Blogger doesn't have a square root sign available, so I will use sqrt(2) to signify the square root of 2, for example.



Here are two sides of a right triangle.  In these problems, c is always the hypotenuse and a and b are the short sides, also called the legs.  Find the missing side using the Pythagorean Theorem, write it as a square root in simplified form and give the approximation to the nearest thousandth.

1)  a = 7, b = 6, c = _________

2) b = 6, c = 7, a = __________

Here are some fractions with the square root in the denominator.  Write them in standard form and simplify.

3) 20/sqrt(10)

4) 15/sqrt(6)


Find the distance between the two given points using the formula Distance = sqrt((x1 - x2)² + (y1 - y2)²). Write the number as a square root in simplified form and give the approximation to the nearest thousandth.

5)  (3, 7) and (-2, 6)

6) (3, 1) and (-5, -9)

Answers in the comments.

Practice with logarithmic scales - Richter and decibel.

a) There was a 7.6 earthquake in Mindanao in the Philippines and a 5.7 aftershock three days later. Rounded to the nearest multiple of 10, how much stronger was the bigger quake?

b) An earthquake registering 5.7 is recorded one morning, and in the afternoon, another quake 27 times stronger is felt. Give the Richter reading of the second stronger quake. (Nearest tenth.)

c) What is the decibel reading of a sound that has 7 times more energy than a 75 decibel reading. (Nearest decibel.)

d) One singer is measured at 64 dB. Eight singers at the same level would have eight times more energy. To the nearest decibel, what is the decibel level of the eight singer chorus?

e)  How much more energy is there in a reading of 75 dB compared to 64 dB.  Round to the nearest whole number.

Answers in the comments.

Practice for underflow and overflow in scientific notation.

Here is a link to a previous post about scientific notation, underflow and overflow.

Consider 80^80.  If you enter this into your calculator, you will likely get an overflow error because the answer is more than 10^100.  Here is how we can get around this, by splitting the number into two parts that are less than 10^100 and multiplying them together.

80^40 = 1.329227996 x 10^76

If we multiply 80^40 by 80^40, we will get 80^80.  We need to square 1.329227996 to get the new significand, and the new exponent will be 10^(76+76) = 10^152

1.32922² = 1.766825808..., but since we only squared the number with six significant digits, we can only trust the answer to five significant digits, so our best answer is 1.7668 x 10^152.


More practice.

a) 40^80

b)40^-80

Answers in the comments.

Changing repeating decimals into fractions in lowest terms.

Using Blogger editing, it's a little tricky to put a bar over a group of numbers, but there is a strikethrough option, so I'm going to use that. To write .16161616...., I'll type .16 and to type .1666666..., I'll type .16.

Sorry if it's a little hard to read.

Method for changing from repeating decimal to fraction in lowest terms.

1. Call the repeating decimal x.
2. Multiply x by a power of ten that has as many zeros as there are digits in the repeating part.
3. Subtract x from the bigger number, which will cancel out the repeating part.
4*. IF the subtraction gives you a decimal number, multiply by some power of ten so you get (whole number times) x = (some other whole number)
5. Divide both sides of the equation by the number multiplying x.
6. Reduce the fraction to lowest terms.

Examples.
Example #1: .16 = 0.1616161616....
1. x = 0.16161616...
2. Because the repeating part has two digits, multiply x by 100 to get 100x = 16.16161616....
3. 100x - x = 16.1616161616... - 0.1616161616...
which reduces to 99x = 16.
step 4 isn't needed.
5. x = 16/99, which is reduced to lowest terms.

Example #2: .16 = 0.166666....
1. x = 0.16666...
2. Because the repeating part has one digit, multiply x by 10 to get 10x = 1.6666....
3. 10x - x = 1.66666... - 0.166666...
which reduces to 9x = 1.5
4. 1.5 isn't a whole number, so multiply by 10 on both sides to get 90x = 15.
5. x = 15/90, which is not in lowest terms.
6. 15/90 = 5/30 = 1/6.

Practice problems.

a) Find the fraction for .23
b) Find the fraction for .23
c) Find the fraction for .234

Answers in the comments.

Practice problems for homework 1 of Summer 2011.

Change these numbers in Hindu-Arabic to Roman numerals.

648 = ______

15,781 = _____

Change these numbers in Roman numerals to Hindu-Arabic.
__
IXCDXLVII = ____

DCXCIV = ____

Find the approximation of 7/37 to the following number of decimal places and find the exact repeating decimal representation.

7/37 to the nearest tenth = _____
7/37 to the nearest hundredth = _____
7/37 to the nearest thousandth = ____

Repeating decimal representation = _____

Find the prime factorization of 48.

Find all the factors of 48.


Answers in the comments.

Problems involving interest, monthly payments and down payments.

a) A house is selling for $300,000. You have $20,000 to put down, and the rest you will be financing on a 30 year loan at 6.25%. What is your monthly payment, rounded to the nearest penny?

b) A house is selling for $300,000. You have $20,000 to put down, and the rest you will be financing on a 15 year loan at 6.25%. What is your monthly payment, rounded to the nearest penny?

c) You have $30,000 to put down and you can afford payments of $1500 a month for mortgage. How expensive a house can you afford to buy on 30 year loan at 6.25% interest?

d) You have $30,000 to put down and you can afford payments of $1500 a month for mortgage. How expensive a house can you afford to buy on 15 year loan at 6.25% interest?

Answers in the comments.

practice for the take home and in class triangle identification problems

Part A) In all the following problems where triangles are defined by lengths, we will use the number 11 and the other numbers must be whole numbers.

obtuse and isosceles: 11, 11, _____
acute and isosceles: 11, 11, _____
obtuse and scalene: 11, ___, ___
right and scalene: 11, ____, ____ (hint: 11 will be one of the legs.)
acute and scalene: 11, ____, ____


Part B) In all the following problems where triangles are defined by angles, we will use one angle of 34° and the other numbers must be whole numbers.

obtuse and isosceles: 34°, ____, _____
acute and isosceles: 34°, ____, _____
obtuse and scalene: 34°, ___, _____ (many correct answers.)
right and scalene: 34°, ____, ____
acute and scalene: 34°, ____, ____ (many correct answers.)

Answers in the comments.

The triangle inequality and Heron's Formula with LOTS of practice problems.

In class, we discussed the Triangle Inequality, a way to check if three given numbers u, v and w, could possibly the lengths of the sides of some triangle. If the numbers are given in any old order, we have two inequalities to check.

u + v >= w
AND
w >= |u - v|

Example: u = 6, v = 12, w = 4

6 + 12 >= 4 Yes, this is true.
AND
4 >= |6 - 12| No, this is false, so these three lengths cannot be the sides of a triangle.

Note: If you make w the long side no matter which way they are handed to you, all you have to check is if u + v >= w.

Let's do another example. u = 10, v = 12 and w = 6.

If we make the long side of 12 be w and the side of 6 be v, then all we have to check is
10 + 6 >= 12, and that is true. These could be the sides of a triangle.

If a triangle is defined for us by the lengths of three sides, the "easiest" way to find the area is Heron's Formula, named for the ancient Greek mathematician Heron, sometimes known as Hero. We add up the three sides to get the perimeter p, then we take ½p and call it s, the semi-perimeter. The area is the square root of the product

s(s - u)(s - v)(s - w).

Let's take the example of sides 6, 10 and 12. The perimeter is 28 and the semi-perimeter is 14. Our formula reads as follows
Area = sqrt(14(14-12)(14-10)(14-6)) = sqrt(14*2*4*8).
Factoring we get sqrt(7*2*2*2*2*2*2*2), which means we have three pairs of twins and one 2 that will be left under the square root sign. This gives us 2*2*2*sqrt(7*2) or 8*sqrt(14). If we want to round to the nearest thousandth 8*sqrt(14) ~= 29.933.

Practice problems . Someone in class asked for a LOT of practice problems so here goes. Here is a list of every possible combination of three positive numbers less than 4. The largest number will be listed last. If u + v = w, then it is a straight line segment with a point on the segment as the vertex. If u + v < w, it's not a triangle at all and only if u + v > w do the three sides create an actual triangle with area.

Part 1. For all these examples, determine if the lengths can create a triangle (answer YES), a straight line (answer STRAIGHT) or not a triangle (answer NO).

a) 1, 1, 1
b) 1, 1, 2
c) 1, 1, 3
d) 1, 1, 4
e) 1, 2, 2
f) 1, 2, 3
g) 1, 2, 4
h) 1, 3, 3
i) 1, 3, 4
j) 1, 4, 4
k) 2, 2, 2
l) 2, 2, 3
m) 2, 2, 4
n) 2, 3, 3
o) 2, 3, 4
p) 2, 4, 4
q) 3, 3, 3
r) 3, 3, 4
s) 3, 4, 4
t) 4, 4, 4

Part 2.
For every triplet of numbers that is a triangle, find the area as a square root and rounded to the nearest thousandth.

Answers in the comments.

Distance, Slope and area of a triangle given two points and the origin (0,0)

The idea of putting horizontal and vertical coordinates on points on a plane is about 400 years old now, and much of the credit goes to mathematician and philosopher René Descartes, working with the amateur mathematician Father Marin Mersenne. The name Cartesian coordinates comes from Descartes' last name. Instead of using a lot of different letters for the coordinates, we will be using subscripts for points, such as (x1, y1), (x2, y2), (x3, y3), etc.

Distance between points: The distance formula is really The Pythagorean Theorem. Our coordinate system makes it easy to create a right triangle between any two points that aren't on the same horizontal or vertical line. (If they are on the same horizontal line, then y1 = y2. If they are on the same vertical line, then x1 = x2.) The horizontal leg of the triangle has length that is the absolute difference |x1 - x2|, and the vertical leg has length |y1 - y2|. The sum of the squares of the legs is the square of the hypotenuse, so we take the square root to find the distance.

Slope: Slope is a number which defines the steepness of a line segment by rise/run, which means the difference in y divided by the difference in x. Any time we have a fraction in a formula, we have to be careful that the denominator is not zero, because we are not allowed to divide by zero. (Quick mnemonic: 0/K is okay because it's zero, N/0 is no can do.) The formula can be written as (y1 - y2)/(x1 - x2) or (y2 - y1)/(x2 - x1). In other words, it doesn't matter if you put the coordinates from point 1 first or the coordinates from point 2 first, AS LONG AS YOU ARE CONSISTENT.

A slope of 0 means the y coordinates are equal and the line is horizontal. A positive slope means the line is "uphill" as we go from left to right and a negative slope means we go "downhill" as we go from left to right. The larger the absolute value of the slope, the steeper a line is. A vertical line has no "run" because the x values are the same, and that means dividing by zero, which we have already said doesn't work. A vertical line has undefined slope. (Sometimes people will say it has no slope, which is technically true but it sounds like the slope is zero. As much as possible, I will used the phrase undefined slope to try to avoid this confusion. If you put "no slope" as an answer when it should be "undefined slope", expect to get a half point marked off.)

The area of a triangle defined by (0, 0), (x1, y1) and (x2, y2): You may have learned that the area of a triangle is 1/2 the base times the height, but sometimes you aren't given the base and the height and other formulas are used instead. If given three coordinate points and one of them is the origin (0, 0), the formula for the area is ½|x1y2 - x2y1|. The absolute value sign is necessary because area, like distance, is usually thought of as non-negative. I write "non-negative" instead of "positive" because if you pick three points at random, it's possible you have three points that are all on the same line, which we call co-linear, and if that happens, what you draw is not really a triangle but instead a line segment, which would mean the base exists but the height is zero, so the area is zero.

Example: Consider the points (7, 1) and (-2, 3).

Distance from (7, 1) and (-2, 3):
sqrt((7 -(-2))²+(1 - 3)²) = sqrt(9² + 2²) = sqrt(81 + 4) = sqrt(85).
(note: 85 is 17x5, so it is a square free number and can't be simplified.)

Slope of the line that runs through (7, 1) and (-2, 3):
(1-3)/(7-(-2)) = -2/9. This means a downhill slope, and every time we move 9 units to the right, we move 2 units down. (Conversly, if we move 9 units to the left, we move 2 units up.)

Note: if we switch the points and write the fraction as (3-1)/(-2-7), we get 2/(-9), which is still -2/9.


Area of the triangle with vertices (0, 0), (7, 1) and (-2, 3).
½|7*3 - 1*-2| = ½|21 -(-2)| = ½|23| = 11½ or 11.5. If we have all points that have integer coordinates (known in the literature as lattice points), then the area will be a whole number or a whole number + ½.

Further problem on triangle area: What if we have three points but none of them is (0, 0)?

In this case, pick one of the points and subtract its x value from all the x values AND subtract its y value from all the y values. What this does is rigidly move the original triangle to one of the exact same size and shape that does have a point at the origin (0, 0). The selection of the point is arbitrary and the area will be the same no matter what point is chosen.

Example: What is the area of the triangle formed by the points (7, 6), (2, -1) and (8, 14)?

I'm going to pick the point (2, -1) to be the point that gets moved to the origin because 2 and -1 are small and easy to subtract from the other values.

(7-2, 6-(-1)) = (5, 7)
(2-2, -1-(-1)) = (0, 0)
(8-2, 14-(-1)) = (6, 15)

The area of the triangle with vertices at (0, 0) , (5, 7) and (6, 15) = ½|5*15 - 7*6| = ½|75 - 42|
= ½|33| = 16.5 or 16½.

Practice problems:

a) Find the distance between (9, 4) and (-1, 2).
b) Find the slope of the line that connects (9, 4) and (-1, 2)
c) Find the area of the triangle with vertices at (0, 0), (9, 4) and (-1, 2).
d) Find the area of the triangle with vertices at (12, 7), (9, 4) and (-1, 2).

Answers in the comments.