Geometry: Transversals and Parallel Lines

Happy Wednesday math friends! In this post we are going to look at parallel lines and transversals and find the oh so many congruent and supplementary angles they form when they come together! Congruent angles that form with these types of lines are more commonly known as Alternate Interior Angles, Alternate Exterior Angles, Corresponding angles, and Supplementary angles. Let’s look at this one step at a time:

What are transversals?

When two parallel lines are cut by a diagonal line ( called a transversal) it looks something like this:

Each angle above has at least one congruent counterpart. There are several different types of congruent relationships that happen when a transversal cuts two parallel lines and we are going to break each down:

1) Alternate Interior Angles:

When a transversal line cuts across two parallel lines, opposite interior angles are congruent.

2) Alternate Exterior Angles:

When a transversal line cuts across two parallel lines, opposite exterior angles are congruent.

3) Corresponding Angles:

When a transversal line cuts across two parallel lines, corresponding angles are congruent.

4) Supplementary Angles:

Supplementary angles are a pair of angles that add to 180 degrees. 180 degrees is the value of distance found within a straight line, which is why you’ll find so many supplementary angles below:

Knowing the different sets of congruent and supplementary angles, we can easily find any missing angle values when faced with the following question:

-> Using our knowledge of congruent and supplementary angles we should be able to figure this out! Right away we can find angle 2 by noticing angle 1 and angle 2 are supplementary angles (add to 180 degrees). 

-> Knowing angle 2 is 50 degrees, we can now fill in the rest of our transversal angles based on our corresponding and supplementary rules.

Try the following transversal and parallel lines questions below! Some may a bit harder than the previous example, if you get stuck, check out the video that goes over a similar example above and happy calculating! 🙂

Practice Questions:

  1. Find the value of the missing angles given line r  is parallel to line  s and line t is a transversal. 

2. Find the value of the missing angles given line r is parallel to line s and line t is a transversal. 

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Algebra 2: Imaginary and Complex Numbers

Happy Wednesday and back to school season math friends! This post introduces imaginary and complex numbers when raised to any power exponent and when multiplied together as a binomial. When it comes to all types of learners, we got you between the video, blog post, and practice problems below. Happy calculating! 🙂

What are Imaginary Numbers?

Imaginary numbers happen when there is a negative under a radical and looks something like this:

Why does this work?

In math, we cannot have a negative under a radical because the number under the square root represents a number times itself, which will always give us a positive number.

Example:

But wait, there’s more:

When raised to a power, imaginary numbers can have the following different values:

Knowing these rules, we can evaluate imaginary numbers, that are raised to any value exponent! Take a look below:

-> We use long division, and divide our exponent value 54, by 4.

-> Now take the value of the remainder, which is 2, and replace our original exponent. Then evaluate the new value of the exponent based on our rules.

What are Complex Numbers?

Complex numbers combine imaginary numbers and real numbers within one expression in a+bi form. For example, (3+2i) is a complex number. Let’s evaluate a binomial multiplying two complex numbers together and see what happens:

-> There are several ways to multiply these complex numbers together. To make it easy, I’m going to show the Box method below:

Try mastering imaginary and complex numbers on your own with the questions below!

Practice:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Algebra: Box Plots, Interquartile Range and Outliers, Explained!

Ahoy math friends! This post takes a look at one method of analyzing data; the box plot method. This method is great for visually identifying outliers and the overall spread of numbers in a data set.

Box plots look something like this:

Screen Shot 2020-09-02 at 11.19.22 AM.png

Why Box Plots?

Box Plots are a great way to visually see the distribution of a set of data.  For example, if we wanted to visualize the wide range of temperatures found in a day in NYC, we would get all of our data (temperatures for the day), and once a box plot was made, we could easily identify the highest and lowest temperatures in relation to its median (median: aka middle number).

From looking at a Box Plot we can also quickly find the Interquartile Range and upper and lower Outliers. Don’t worry,  we’ll go over each of these later, but first, let’s construct our Box Plot!

Screen Shot 2020-09-02 at 11.20.42 AM->  First, we want to put all of our temperatures in order from smallest to largest.
Screen Shot 2020-09-02 at 11.21.28 AM.png-> Now we can find Quartile 1 (Q1), Quartile 2 (Q2) (which is also the median), and Quartile 3 (Q3).  We do this by splitting the data into sections and finding the middle value of each section.Screen Shot 2020-09-05 at 11.19.22 PM

Q1=Median of first half of data

Q2=Median of entire data set

Q3=Median of second half of data

-> Now that we have all of our quartiles, we can make our Box Plot! Something we also have to take notice of, is the minimum and maximum values of our data, which are 65 and 92 respectively. Let’s lay out all of our data below and then build our box plot:

Screen Shot 2020-09-05 at 11.19.27 PM

Screen Shot 2020-09-05 at 11.20.45 PM

Now that we have our Box Plot, we can easily find the Interquartile Range and upper/lower Outliers.

Screen Shot 2020-09-05 at 11.21.54 PM

->The Interquartile Range is the difference between Q3 and Q1. Since we know both of these values, this should be easy!

Screen Shot 2020-09-05 at 11.22.02 PMNext, we calculate the upper/lower Outliers.

Screen Shot 2020-09-05 at 11.23.45 PM

-> The Upper/Lower Outliers are extreme data points that can skew the data affecting the distribution and our impression of the numbers. To see if there are any outliers in our data we use the following formulas for extreme data points below and above the central data points.

Screen Shot 2020-09-05 at 11.24.27 PM*These numbers tell us if there are any data points below 44.75 or above 114.75, these temperatures would be considered outliers, ultimately skewing our data. For example, if we had a temperature of  Screen Shot 2020-09-05 at 11.26.38 PMor Screen Shot 2020-09-05 at 11.29.25 PM these would both be considered outliers.

Screen Shot 2020-09-05 at 11.24.35 PM

Practice Questions:

Screen Shot 2020-09-05 at 11.34.21 PMSolutions:

Screen Shot 2020-09-05 at 11.37.06 PM

Screen Shot 2020-09-05 at 11.37.39 PM

Screen Shot 2020-09-05 at 11.38.10 PM

Screen Shot 2020-09-05 at 11.39.06 PM

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Geometry: How to Construct an Equilateral Triangle?

 

Happy Wednesday math peeps! This post introduces constructions by showing how to construct an equilateral triangle by using a compass and a ruler. For anyone new to constructions, this is the perfect topic for art aficionados since there is more drawing than there is actual math.  Screen Shot 2020-08-25 at 4.09.58 PM.pngEquilateral Triangle: A triangle with three equal sides.  Not an easy one to forget, the equilateral triangle is super easy to construct given the right tools (compass+ ruler). Take a look below:Screen Shot 2020-08-25 at 3.56.17 PM.png

Solution:

Construction-GIF-v2

What’s Happening in this GIF? 

1. Using a compass, measure out the distance of line segment  Screen Shot 2020-08-25 at 4.19.02 PM.

 2. With the compass on point A, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

 3. With the compass on point B, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

4. Notice where the arcs intersect? Using a ruler, connect points A and B to the new point of intersection. This will create two new equal sides of our triangle!

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

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Algebra 2: Expanding Cubed Binomials

Greetings math friends! This post will go over how to expand and simplify cubed binomials 2 different ways. We’re so used to seeing squared binomials such as, Screen Shot 2020-08-19 at 11.29.14 AM.png, and expanding them without a second thought.  But what happens when our reliable squared binomials are now raised to the third power, such as,Screen Shot 2020-08-19 at 11.29.48 AM?  Luckily for us, there is a Rule we can use:

Screen Shot 2020-08-18 at 10.12.33 PM

But where did this rule come from?  And how can we so blindly trust it? Which is why we are going to prove the above rule here and now using 2 different methods:Screen Shot 2020-08-19 at 11.31.13 AM

Why bother? Proving this rule will allow us to expand and simplify any cubic binomial given to us in the future! And since we are proving it 2 different ways, you can choose the method that best works for you.

Method #1: The Box MethodScreen Shot 2020-08-18 at 10.14.37 PMScreen Shot 2020-08-18 at 10.14.55 PM.pngScreen Shot 2020-08-18 at 10.15.06 PMScreen Shot 2020-08-18 at 10.15.39 PM.pngScreen Shot 2020-08-18 at 10.15.50 PM

Screen Shot 2020-08-19 at 2.24.54 PMScreen Shot 2020-08-19 at 2.53.43 PM

Screen Shot 2020-08-19 at 2.29.22 PM.pngScreen Shot 2020-08-18 at 10.17.19 PM.png

Screen Shot 2020-08-19 at 2.27.56 PMScreen Shot 2020-08-19 at 2.54.36 PM.png

Screen Shot 2020-08-18 at 10.21.05 PM.png

Method #2: The Distribution MethodScreen Shot 2020-08-18 at 10.17.54 PM.pngScreen Shot 2020-08-18 at 10.19.49 PMScreen Shot 2020-08-19 at 2.42.11 PM

Screen Shot 2020-08-18 at 10.21.05 PM.png

Now that we’ve gone over 2 different methods of cubic binomial expansion, try the following practice questions on your own using your favorite method!

Practice Questions: Expand and simplify the following.

Screen Shot 2020-08-18 at 10.21.56 PM

Solutions:

Screen Shot 2020-08-18 at 10.22.19 PM.png

Still, got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

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**Bonus: Test your skills with this Regents question on Binomial Cubic Expansion!

 

Algebra: How to use Recursive Formulas?

Welcome to Mathsux! This post is going to show you everything you need to know about Recursive Formulas by looking at three different examples. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page.  Happy calculating! 🙂

What is a Recursive Formula?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value.  What does that mean?  Check out the example below for a clearer picture:

Example #1:

Screen Shot 2020-08-11 at 8.12.21 AM.pngScreen Shot 2020-08-11 at 8.12.33 AMScreen Shot 2020-08-11 at 9.18.18 AM

Screen Shot 2020-08-11 at 8.13.07 AMScreen Shot 2020-08-11 at 8.13.36 AM.pngScreen Shot 2020-08-11 at 9.21.01 AM.pngScreen Shot 2020-08-11 at 8.14.34 AM.pngScreen Shot 2020-08-11 at 8.14.49 AMExample #2:

Screen Shot 2020-08-11 at 8.15.19 AM.png

Screen Shot 2020-08-11 at 8.15.38 AMScreen Shot 2020-08-11 at 9.22.10 AMScreen Shot 2020-08-11 at 8.52.36 AMScreen Shot 2020-08-11 at 8.52.52 AM.pngScreen Shot 2020-08-11 at 9.23.54 AM.png

***Note this was written in a different notation but is solved in the exact same way!

Screen Shot 2020-08-11 at 8.53.24 AM.pngScreen Shot 2020-08-11 at 8.53.35 AM

Example #3:Screen Shot 2020-08-11 at 8.54.05 AM.pngScreen Shot 2020-08-11 at 8.54.19 AMScreen Shot 2020-08-11 at 9.24.42 AMScreen Shot 2020-08-11 at 8.54.49 AMScreen Shot 2020-08-11 at 9.25.41 AM.pngScreen Shot 2020-08-11 at 8.56.22 AMScreen Shot 2020-08-11 at 8.56.36 AM.pngPractice Questions:

Screen Shot 2020-08-11 at 10.04.21 AM

Solutions:

Screen Shot 2020-08-11 at 9.02.18 AM.png

Still got questions? No problem! Check out the video above for more or try the NYS Regents question below, and please don’t hesitate to comment with any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas?  Click here!

Geometry: Reflections

Greetings and welcome to Mathsux! Today we are going to go over reflections, one of the many types of transformations that come up in geometry.  And thankfully, it is one of the easiest transformation types to master, especially if you’re more of a visual learner/artistic type person. So let’s get to it!

What are Reflections?

Reflections on a coordinate plane are exactly what you think! When a point, a line segment, or a shape is reflected over a line it creates a mirror image.  Think the wings of a butterfly, a page being folded in half, or anywhere else where there is perfect symmetry.

Check out the Example below:

Screen Shot 2020-08-04 at 5.19.40 PM

Screen Shot 2020-08-04 at 4.57.07 PMScreen Shot 2020-08-04 at 4.57.34 PM.pngScreen Shot 2020-08-04 at 4.57.55 PMScreen Shot 2020-08-04 at 4.58.10 PM.pngScreen Shot 2020-08-04 at 4.59.19 PMScreen Shot 2020-08-04 at 4.59.36 PM.pngScreen Shot 2020-08-05 at 9.16.37 AM.png
Screen Shot 2020-08-04 at 5.00.19 PM.pngScreen Shot 2020-08-04 at 5.00.43 PMScreen Shot 2020-08-04 at 5.01.02 PM.png

Practice Questions:

Screen Shot 2020-08-04 at 5.13.52 PM

Solutions:

Screen Shot 2020-08-04 at 5.15.37 PM.png

Still got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

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Geometry: Intersecting Secant Theorem

Ahoy! Today we’re going to cover the Intersecting Secants Theorem!  If you forgot what a secant is in the first place, don’t worry because all it is a line that goes through a circle.  Not so scary right? I was never scared of lines that go through circles before, no reason to start now.

If you have any questions about anything here, don’t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! 🙂

Wait, what are Secants?

Screen Shot 2020-07-14 at 10.07.54 PM

Intersecting Secants Theorem: When secants intersect an amazing thing happens! Their line segments are in proportion, meaning we can use something called the Intersecting Secants Theorem to find missing line segments.  Check it out below: 

Screen Shot 2020-07-14 at 10.44.53 PM

Let’s now see how we can apply the intersecting Secants Theorem to find missing length.

Screen Shot 2020-07-14 at 10.45.29 PM.png

Screen Shot 2020-07-14 at 10.10.23 PMScreen Shot 2020-07-14 at 10.10.39 PM.pngScreen Shot 2020-07-14 at 10.11.13 PMScreen Shot 2020-07-14 at 10.11.52 PM.pngScreen Shot 2020-07-14 at 10.13.24 PMScreen Shot 2020-07-14 at 10.13.57 PM.pngScreen Shot 2020-07-14 at 10.14.20 PM

Screen Shot 2020-07-14 at 10.14.41 PM.png

Ready to try the practice problems below on your own!?

Practice Questions: Find the value of the missing line segments x.

Screen Shot 2020-07-14 at 10.38.02 PM

Screen Shot 2020-07-20 at 9.30.01 AM

Solutions:

Screen Shot 2020-07-20 at 9.30.55 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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To review a similar NYS Regents question check out this post here

Algebra: Absolute Value Equations

Happy Wednesday math friends! Today, we’re going to go over how to solve absolute value equations.  Solving for absolute value equations supplies us with the magic of two potential answers since absolute value is measured by the distance from zero.  And if this sounds confusing, fear not, because everything is explained below!

Also, if you have any questions about anything here, don’t hesitate to comment. Happy calculating! 🙂

Absolute Value measures the “absolute value” or absolute distance from zero.  For example, the absolute value of 4 is 4 and the absolute value of -4 is also 4.  Take a look at the number line below for a clearer picture:

Screen Shot 2020-07-08 at 2.02.40 PM.png

Now let’s see how we can apply our knowledge of absolute value equations when there is a missing variable!Screen Shot 2020-07-08 at 2.03.07 PMScreen Shot 2020-07-08 at 2.03.46 PM.pngScreen Shot 2020-07-08 at 2.04.00 PMScreen Shot 2020-07-08 at 2.04.26 PM.pngScreen Shot 2020-07-08 at 2.04.56 PM

Screen Shot 2020-07-08 at 2.05.17 PM.png

Screen Shot 2020-07-08 at 2.05.39 PMNow let’s look at a slightly different example:

Screen Shot 2020-07-11 at 4.49.57 PM.pngScreen Shot 2020-07-08 at 2.07.59 PM

Screen Shot 2020-07-08 at 2.07.41 PM.png

Screen Shot 2020-07-08 at 2.08.26 PM.png

Screen Shot 2020-07-08 at 2.08.46 PM

Screen Shot 2020-07-08 at 2.09.33 PMScreen Shot 2020-07-08 at 2.09.58 PM.png Screen Shot 2020-07-08 at 2.10.39 PM.pngScreen Shot 2020-07-08 at 2.10.50 PM

Practice Questions: Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Screen Shot 2020-07-16 at 9.01.08 AM.png

 

 

 

 

 

Solutions:

Screen Shot 2020-07-08 at 2.12.04 PM

Still got questions?  No problem! Check out the video the same examples outlined above. Happy calculating! 🙂

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