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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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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Algebra: How to Graph y=mx+b

Hi everyone, welcome back to Mathsux! This week we’ll be reviewing how to graph an equation of a line in y=mx+b form. And if you have not checked out the video below, please do! Happy calculating! 🙂

 

Graphing an Equation of Line: An equation of a line can be represented by the formula:Screen Shot 2020-06-17 at 9.07.16 PM

Y-Intercept: This is represented by b, the stand-alone number in y=mx+b. This represents where the line hits the y-axis.  This is always the first point you want to start with when graphing at coordinate point (0,b).

Slope: This is represented by m, the number next to x in y=mx+b. Slope tells us how much we go up or down the y-axis and left or right on the x- axis in fraction form:

Screen Shot 2020-06-17 at 9.09.42 PM

Now let’s check out an Example!

Graph the equation of a line Screen Shot 2020-06-17 at 9.10.42 PM.

Screen Shot 2020-06-17 at 9.12.01 PM

Screen Shot 2020-06-17 at 9.12.35 PM

Screen Shot 2020-06-17 at 9.13.34 PM

Screen Shot 2020-06-17 at 9.14.20 PMTry the following practice questions on your own!

Practice Questions:

Screen Shot 2020-06-17 at 9.15.22 PM

Screen Shot 2020-06-17 at 9.16.21 PM

Want more Mathsux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

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Geometry: Perpendicular and Parallel Line Through a Given Point

Happy Wednesday math friends! Today we’re going to go over the difference between perpendicular and parallel lines. Then we’ll use our knowledge of equation of a line (y=mx+b) to see how to find perpendicular and parallel lines through a given point.  This is is a common question that comes up on the NYS Geometry Regents and is something we should prepare for, so let’s go!

If you need any further explanation, don’t hesitate to check out the Youtube video below that goes into detail on how to solve these types of questions one step at a time. Happy calculating! 🙂

Perpendicular lines: Lines that intersect to create a 90-degree angle and can look something like the graph below.  Their slopes are negative reciprocals of each other which means they are flipped and negated. See below for example!Screen Shot 2020-06-10 at 10.26.10 AM

Example: Find an equation of a line that passes through the point (1,3) and is perpendicular to line y=2x+1 .

Screen Shot 2020-06-10 at 10.28.20 AMScreen Shot 2020-06-10 at 10.27.43 AMScreen Shot 2020-06-10 at 10.28.42 AMScreen Shot 2020-06-10 at 10.29.06 AM

Parallel lines are lines that go in the same direction and have the same slope (but have different y-intercepts). Check out the example below!

Screen Shot 2020-06-10 at 10.29.26 AMExample: Find an equation of a line that goes through the point (-5,1) and is parallel to line y=4x+2.Screen Shot 2020-06-10 at 10.34.46 AMScreen Shot 2020-06-10 at 10.35.23 AMTry the following practice questions on your own!

Practice Questions:

1) Find an equation of a line that passes through the point (2,5) and is perpendicular to line y=2x+1.

 2) Find an equation of a line that goes through the point (-2,4) and is perpendicular to lineScreen Shot 2020-06-10 at 11.24.06 AM

 3)  Find an equation of a line that goes through the point (1,6) and is parallel to line y=3x+2.

4)  Find an equation of a line that goes through the point (-2,-2)  and is parallel to line y=2x+1.

Solutions:Screen Shot 2020-06-10 at 11.22.05 AM

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! 🙂

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Algebra: 4 Ways to Factor Quadratic Equations

*If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-06-02 at 3.03.55 PMScreen Shot 2020-06-02 at 3.04.24 PM____________________________________________________________________Screen Shot 2020-06-02 at 3.20.04 PM____________________________________________________________________

Screen Shot 2020-06-02 at 3.07.02 PM____________________________________________________________________
Screen Shot 2020-06-02 at 3.07.42 PM
Choose the factoring method that works best for you and try the practice problems on your own below!

Practice Questions:

Screen Shot 2020-06-02 at 3.09.58 PMSolutions:

Screen Shot 2020-06-02 at 3.10.30 PM

Want a review of all the different factoring methods out there?  Check out the ones left out here (DOTS and GCF) and happy calculating! 🙂

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Geometry: Median of a Trapezoid Theorem

*If you haven’t done so, check out the video that goes over this exact problem, also please don’t forget to subscribe!

Medians of a Trapezoid copy

Screen Shot 2020-06-02 at 7.31.07 AMStep 1:  Let’s apply the Median of a Trapezoid Theorem to this question!  A little rusty?  No problem, check out the Theorem below.

Median of a Trapezoid Theorem: The median of a trapezoid is equal to the sum of both bases.Screen Shot 2020-06-02 at 7.32.31 AMStep 2: Now that we found the value of x , we can plug it back into the equation for Screen Shot 2020-06-02 at 7.33.44 AMmedian,  to find the value of median Screen Shot 2020-06-02 at 7.34.25 AM

Screen Shot 2020-06-02 at 7.34.48 AM

Want more practice?  Your wish is my command! Check out the practice problems below:

Practice Questions:

1.Screen Shot 2020-06-02 at 7.35.29 AMis the median of trapezoid ABCDEF, find the value of the median, given the following:Screen Shot 2020-06-02 at 7.35.47 AM2. Screen Shot 2020-06-02 at 9.01.08 AMis the median of trapezoid ACTIVE, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.16.22 AM3.Screen Shot 2020-06-02 at 9.17.01 AMis the median of  trapezoid DRAGON, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.22.13 AM

4. Screen Shot 2020-06-02 at 9.23.08 AMis the median of trapezoid MATRIX, find the value of the median, given the following:Screen Shot 2020-06-02 at 9.23.43 AM

Solutions:

Screen Shot 2020-06-02 at 9.25.05 AM

Need more of an explanation?  Check out the detailed video and practice problems. Happy calculating! 🙂

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Geometry: Area of a Sector

Youtube Area of a sector copy

Hi math friends, has anyone been cooking more during quarantine?  We all know there is more time for cookin’ and eatin’ cakes but have you ever been curious about the exact amount of cake you are actually eating?! Well, you’re in luck because today we are going to go over how to find the area of a piece of cake, otherwise known as the Area of a Sector!

Now, we’ll all be able to calculate just how much we are overdoing it on that pie! Hopefully, everyone is eating better than I am (I should really calm down on the cupcakes).  Ok, now to our question:

*Also, If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-05-19 at 4.18.42 PMExplanation:

How do I answer this question? 

We must apply/adjust the formula for the area of a circle to find the area of the blue shaded region otherwise known as the sector of this circle.                                                    

How do we do this?    

Before we begin let’s review the formula for the area of a circle. Just a quick reminder of what each piece of the formula represents:

Screen Shot 2020-05-19 at 4.24.28 PMStep 1: Now let’s fill in our formula, we know the radius is 5, so let’s fill that in below:

Screen Shot 2020-05-19 at 4.26.08 PMStep 2: Ok, great! But wait, this is for a sector; We need only a piece of the circle, not the whole thing.  In other words, we need a fraction of the circle. How can we represent the area of the shaded region as a fraction?

Well, we can use the given central angle value, Screen Shot 2020-05-19 at 4.27.17 PM, and place it over the whole value of the circle,Screen Shot 2020-05-21 at 4.01.12 PM . Then multiply that by the area of the entire circle. This will give us the value we are looking for!

Screen Shot 2020-05-19 at 4.27.45 PMStep 3: Multiply and solve!Screen Shot 2020-05-19 at 4.28.38 PM

Ready for more? Try solving these next few examples on your own to truly master area of a sector!

Find the area of each shaded region given the central angle and radius for each circle:Screen Shot 2020-05-19 at 4.29.40 PM

Check the solutions below, when you’re ready:Screen Shot 2020-05-19 at 4.30.36 PMWhat do you think of finding the area of sector? Are you going to measure the area of your next slice of pizza?  Do you have any recipes to recommend?  Let me know in the comments and happy calculating! 🙂