How to Construct the Altitudes of a Triangle ⊿

Hi everyone and welcome back to another week of MathSux! For this week’s math lesson, I bring to you four constructions in one in How to Construct the Altitudes of a Triangle. In the video below, we will:

  • Define what an altitude is
  • Find the altitude of an acute, obtuse, and right triangles
  • Discover and define the orthocenter

I hope you find the below video helpful and interesting! Let me know in the comments below and happy calculating!

How to Construct the Altitudes of a Triangle :

An Altitude is a perpendicular line drawn from the vertex of a triangle to the opposite side, creating a 90º angle.

How to Construct the Altitudes of a Triangle

In the video above, we will look at how to find the altitude of an acute obtuse, and right triangle.

How to Find the Orthocenter of Triangle with a Compass:

The Orthocenter is a point where all three altitudes meet within a triangle.

How to Construct the Altitudes of a Triangle

In order to find the orthocenter using a compass, all we need to do is find the altitude of each vertex. The point at which they meet is the orthocenter. Check out the video above to see how this works step by step.

More Constructions?

If you’re looking for more constructions out there, be sure to check out the MathSux Constructions Playlist on YouTube for even more tutorials! I hope you find this video helpful and hope it motivates you to learn even more about constructions (or you know get through your homework/pass that test). Whatever reason you have to learn about constructions I hope this helps! Stay positive and happy calculating!

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

Facebook ~ Twitter ~ TikTok ~ Youtube

Check out more Geometry posts and Constructions step by step here!

Derivatives Practice Questions

Hi everyone and welcome to MathSux! In this weeks post, we will venture into Calculus for the first time! I won’t get too much into the nitty gritty explanation of what derivative are here, but instead will provide a nice overview of Derivatives Practice Questions. This post includes everything you need to know about finding the derivatives of a function including the Power Rule, Product Rule, Quotient Rule, and the Chain Rule. Below you will see examples, a Derivative Rules Cheat Sheet, and of course practice questions! I hope these quick examples help in the classroom or for that test coming up! Let me know if it helps and you want more Calculus lessons. Happy Calculating!

What is a Derivative?

We use the derivative to find the rate of change of a function with respect to a variable. You can find out more about what a derivative is and its proper notation here at mathisfun.com. Read on below for a derivative rules cheat sheet, examples, and practice problems!

Derivative Rules Cheat Sheet:

Power Rule:

The power rule is used for finding the derivative of functions that contain variables with real exponents.  Note that the derivative of any lone constant number is zero.

derivatives of functions with exponents

Product Rule:

The product rule is used to find the derivative of two functions that are being multiplied together.

derivatives practice questions

Quotient Rule:

Applying the quotient rule, will find the derivative of any two functions set up as a ratio.  Be sure to notice any numbers or variables in the denominator that can be brought to the numerator (if that’s the case, can use the more friendly power rule).

derivatives of trigonometric functions

Chain Rule:

The chain rule allows us to find the derivative of nested functions. This is great for trigonometric functions and entire functions that are raised to an exponent.

calculus chain rule examples

Ready for some practice questions!? Check out the ones below to test your knowledge of derivatives!

Derivatives Practice Questions:

Find the derivatives for each function below.

Derivatives Practice Questions

Solutions:

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

*Also, if you want to check out Rate of Change basics click this link here!

Facebook ~ Twitter ~ TikTok ~ Youtube

NYS Regents Review – Algebra June 2021

Greetings math friends! Today we are going to break down the NYS Regents, specifically the Algebra NYS Common Core Regents from June 2021, one question at a time. The following video playlist goes over each and every question one step at a time. I’ve been working on this playlist slowly adding new questions and videos every week and now that it is complete, it is time to celebrate (and/or study)! Please enjoy this review along with the study aids and related links that will also help you ace the NYS Regents. Happy calculating!

NYS Regents

NYS Regents – Algebra June 2021 Playlist

Study Resources:

Looking to ace your upcoming NYS Regents!? Don;t forget to check out the resources below, including an Algebra Cheat Sheet, and important topics and videos to review. Good luck and happy calculating!😅

Algebra Cheat Sheet

Combing Like Terms and the Distributive Property

How to Graph an Equation of a Line

Piecewise Functions

NYS Regents – Algebra 2020

How to Study Math?

How is one supposed to study math!? Well, there is usually only one way, and that is to practice, practice, practice! But don’t get too stressed, because you can also make practice fun (or at least more pleasant).

Add some background music to your study session and make a nice cup of tea before diving in for the brain marathon. Another idea is to study only 1 hour at a time and to be sure to take breaks. Can’t seem to get that one question? Take a break and walk a way, or even better find a new study spot! It’s been scientifically proven that studying in different places can boost your memory of the very information you’re trying to understand.

What study habits do you have that have worked for you in the past? Let me know in the comments and good luck on your upcoming test!

Facebook ~ Twitter ~ TikTok ~ Youtube

** Be sure to check out the full list of Algebra lessons and old Regents questions review!**

Legs of a Right Triangle (when an altitude is drawn)

Greetings math peeps and welcome to another week of MathSux! In todays post we are going to explore how to find the legs of a right triangle when an altitude is drawn from the vertex to its hypotenuse. We are going to take this step by step on how to solve a problem like this. Although, I go over the long way to solve this problem, there is also short cut many people use called the “geometric means” which is also briefly mentioned in this post (under the Tip! section). Whichever method you choose, do what makes most sense to you! Happy calculating!

Similar Right Triangles (with Altitude drawn):

When two triangles have equal angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but if they have the same angles and the sides are in proportion, they are similar! Triangles can be proven similar by AA, SAS, or SSS. For more on similar triangles, check out this post here.

There is a special type of scenario that happens with similar right triangles. When an altitude is drawn from the vertex of a right triangle, it forms two smaller triangles, which creates three right triangles that are similar to the original triangle, based on Angle Angle (AA). Check out the example to see how it works!

In triangle ABC, an altitude is drawn from angle A to its hypotenuse BC. Notice that this creates three 90º right triangles in total (ABC, ABD, and ADC). 

Drawing an altitude created three total right triangles, broken out below:

If you take a closer look, at the triangles above you’ll notice that these two new triangles (ABD and ADC) share an angle with the original bigger triangle ABC and a 90º angle. This makes each of the new triangles similar to the original triangle by AA. We can therefore say that:

How are the New Right Triangles Similar?

1) Triangle ABC ~ ADC

2) Triangle ABC ~ ADB

Ready for an Example?  Lets check out the one below!

Legs of a Right Triangle

Step 1: To find the length of the missing legs of a right triangle, first, let’s separate each right triangle to see what we’re working with, along with the values of each length that was provided.

Legs of a Right Triangle

Step 2: We want to find the length of side AC, so let’s use the triangles whose sides include AC.  This leads us to use triangles ABC and ADC.

Legs of a Right Triangle

Step 3:  Now, let’s set up our proportion to find our missing side.  To make it easier for ourselves let’s first flip and rotate triangle ADC to line up with triangle ABC.

Legs of a Right Triangle
Legs of a Right Triangle

Think you’re ready to try some practice questions on your own? Check out the ones below!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating!

Facebook ~ Twitter ~ TikTok ~ Youtube

Check out more posts on Similar Triangles here!

Sum of Infinite Geometric Sequence

Hey math friends and happy Wednesday! Today we are going to take a look at how to find the sum of infinite geometric sequence (aka series) in summation notation. This may sound complicated, but lucky for us there is an already existing formula that is ready and easy for us to use! So, let’s get to it! Also, don’t forget to check out the video and practice problems below for even more. Happy calculating! 🙂

What does it mean to find the “Sum of Infinite Geometric Sequence” (Series)?

We already know what a geometric sequence is: a sequence of numbers that forms a pattern when the same number is multiplied or divided to each term.

Example:

But when what happens if we wanted to sum all the terms of our geometric sequence together?

Example:

We can also write our infinite geometric series using Summation Notation:

How would we calculate that?  That’s where our Infinite Geometric Series formula will come in handy! But before we find a solution, lets take a closer look at what geometric series qualify for this formula.

Divergence Vs. Convergence:

There are two types of Infinite Geometric Series:

Type 1: Infinite Geometric Series that diverges to infinity

Type 2: Infinite Geometric Series that converges to a numeric value (-1 < r < 1)

Check out the differences in the example below:

Infinite Geometric Series Formula:

Note that the below infinite geometric series formula can only be used if the common ratio, r, is less than 1 and greater than -1. If the common ratio, r, is not between -1 and 1, then the sum of the geometric sequence diverges to infinity (and the formula cannot be used).

Now that we have a formula to work with and know when to use it (when -1 < r < 1), let’s take another look at our question and apply our infinite geometric series formula to find a solution:

Sum of Infinite Geometric Sequence

Step 1: First, let’s identify the common ratio to make sure that its between -1 and 1.

Sum of Infinite Geometric Sequence

Step 2: Now that we know we can use our formula, let’s write out each part and identify what numbers we are going to plug in.

Sum of Infinite Geometric Sequence

Step 3: Now let’s fill in our formula and solve with the given values.

Sum of Infinite Geometric Sequence

Practice Questions:

State if each series converges or diverges, then if applicable find the solution.

Sum of Infinite Geometric Sequence

Solutions:

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

*Also, if you want to check out Finite Geometric Series click this link here!

Facebook ~ Twitter ~ TikTok ~ Youtube

Back to School Review: Algebra | Geometry | Algebra 2/Trig.

Back to School Review

Hey math friends and welcome back to MathSux! Back to school season is upon us which means most students (and teachers) will need to review a bit before diving into a completely new subject.  In order to alleviate some of the back to school whoas, I bring to you, this back to school review! Check out the videos below to get the math juices flowing whether you’re new to Algebra, Geometry, or Algebra 2/Trig! I hope you find these videos helpful and wish everyone the best of luck in their first days at school! Happy calculating! 🙂

How to Prepare for Algebra: 

Calling all incoming algebra students, Combining Like Terms is a great place to start! You most likely have combined like terms before, but there’s nothing like sharpening your skills before getting the intense Algebra questions that are coming your way. Check out the video below and try the practice questions here!

Practice Problems: https://mathsux.org/2020/09/30/algebra-combining-like-terms-and-distributive-property/

How to Prepare for Geometry: 

Geometry students, you have the world of shapes ahead of you! It’s an exciting time to review basic Area, Perimeter, Circumference, and Pythagorean Theorem rules before moving ahead with this subject. Review the Pythagorean Theorem below from Khan Academy and check out the last page of the review sheet here to review area and perimeter.

How to Prepare for Algebra 2:

Relieve the fond memories of algebra by reviewing all the different ways to Factor and Solve Quadratic Equations! This is a great way to prepare for rational expressions and the harder algebra 2 problems that are right around the corner.  Check out the video below and related practice questions here to reinforce these hopefully not yet forgotten algebra skills!

Practice Problems: https://mathsux.org/2020/06/09/algebra-4-ways-to-factor-trinomials/
Practice Problems: https://mathsux.org/2016/07/06/algebra-2-factor-by-grouping/

Hope you find this quick review helpful before diving in for the real deal! Besides brushing up on these math topics, what type of new school year routines do like to practice in your classroom or at home? Let me know in the comments and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Olympics Statistics: Top 10 Medals by Country

Olympics Statistics

Greetings and welcome back to MathSux! This week, in honor of the Tokyo Olympics, I will be breaking down some Olympic Statistics. We will look at the top 10 countries that hold the most medals and then look at the top 10 medals earned by country in relation to each country’s total population. Let’s take a look and see what we find! Also, please note that all data used for this analysis was found on the website, here. Anyone else watching the Olympics? Try downloading the data with the link above and see what type of conclusions you can find! Happy Calculating! 🙂

Top 10 Countries: Total Olympic Medals

Below shows the top 10 total medals earned by country from the beginning of the Olympics in 1896 to present day July 2021. As we can see in the graph below, the United States is way ahead of the game with thousands more Olympic medals when compared to any other country in the entire world! I always knew the U.S. did well in the Olympics, but did not realize it was to this magnitude!

Top 10 Countries: Total Olympic Medals Based on Population

Below is a different kind of graph. This percentage rate represents total medals earned over time from 1896 to July 2021 divided by the country’s total population. In this case, we can see that Lichtenstein has earned way more medals based on their small population size when compared to any other country in the world! This is amazing and unexpected!

Remember that all data for the above graphs were made from the following website, here. Are you surprised by the above graphs and conclusions? Try downloading the data on your own and see what you can conclude using your own Olympics Statistics skills! Happy calculating! 🙂

Looking to apply more math to the real world? Check out how to find volume of the Hudson Yards Vessal in NYC here

Facebook ~ Twitter ~ TikTok ~ Youtube

How to Construct a Perpendicular Line through a Point on the Line

Greetings math peeps and welcome to another week of MathSux! In this post, we will learn how to construct a perpendicular line through a point on the line step by step. In the past, we learned how to construct a perpendicular bisector right down the middle of the line, but in this case we will learn how to create a perpendicular line through a given point on the line (which is not always in the middle). Following along with the GIF or check out the vide below. Thanks for stopping by and happy calculating! 🙂

What are Perpendicular Lines ?

  • Lines that intersect to create four 90º angles about the two lines.
How to Construct a Perpendicular Line through a Point on the Line

What is happening in this GIF?

Step 1: First, we are going to gather materials, for this construction we will need a compass, straight edge, and markers.

Step 2: Notice that we need to make a perpendicular line going through point B that is given on our line.

Step 3: Open up our compass to any distance (something preferably short though to fit around our point and on the line).

Step 4: Place the compass end-point on Point B, and draw a semi-circle around our point, making sure to intersect the given line.

Step 5: Open up the compass (any size) and take the point of the compass to the intersection of our semi-circle and given line.  Then swing our compass above the line.

Step 6: Keeping that same length of the compass, go to the other side of our point, where the given line and semi-circle connect.  Swing the compass above the line so it intersects with the arc we made in the previous step.

Step 7: Mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line going through Point B!

Still got questions? No problem! Don’t hesitate to comment with any questions below or check out the video above. Thanks for stopping by and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Want to see how to construct a square inscribed in a circle? Or maybe you want to construct an equilateral triangle? Click on each link to view each construction!

Origami and Volume of a Box and Square Base Pyramid

Greetings and happy summer math peeps! In honor of the warm weather and lack of school, I thought we’d have a bit of fun with origami and volume! In this post, we will find the volume of a box and the volume of a square base pyramid. We will also be creating each shape by using origami and following along with the video below. For anyone who wants to follow along with paper folding tutorial, please note that we will need one piece of printer paper that is 8.5″ x 11″and one piece of square origami paper that is 8″ x 8″. If you’re interested in more math and art projects check out this link here. Stay cool and happy calculating! 🙂

Volume of Box (or Rectangular Prism):

To get the volume of our origami box (video tutorial above), we are going to multiply the length times the width times the height. All the values and units of measurement were found by measuring the box we made in inches in the video above with 8.5 x 11 inch computer paper.

Origami and Volume
Origami and Volume

Volume of Square Base Pyramid:

Below is a diagram of the square base pyramid we created via paper folding (watch video tutorial above to follow along!). Please note that if you used a different sized paper (other than 8 X 8 inches), you will get a different value for measurements and for volume.

Origami and Volume
Origami and Volume

For step by step instruction, don’t forget to check out the video above to see how to paper fold a box and square base pyramid. I hope this post made math suck just a little bit less and finding volume a bit more fun. Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

For more Math + Art, check out this post on Perspective Drawing here.

The Original Spirograph: Math + Art

Happy Summer everyone! Now that school is out, I thought we could have a bit of fun with Math and Art! In this post, we will go over how to make a the original spirograph (by hand) step by step using a compass and straight edge. Follow along with the video below or check out the tutorial in pictures in this post. Hope everyone is off to a great summer. Happy calculating! 🙂

What is a Spirograph?

The childhood toy we all know and love was invented by Denys Fisher, a British Engineer in the 1960’s.But the method of creating Spirograph patterns was invented way earlier by engineers and mathematicians in the 1800’s.

The Original Spirograph (by hand):

The Original Spirograph

Step 1: Gather materials, for this drawing, we will need a compass and straight edge.

The Original Spirograph

Step 2: Using our compass, we are going to open it to 7 cm and draw a circle.

The Original Spirograph

Step 3: Next, we are going to open the compass to 1cm, making marks all around the circle, keeping that same distance on the compass.

The Original Spirograph

Step 4: Draw a line connecting two points together (any two points some distance apart will do).

The Original Spirograph

Step 5: Now, we are going to move the straight edge forward by one point each and connect the two points with another line.

The Original Spirograph

Step 6: Continue this pattern of moving the ruler forward by one point and connecting them together all the way around.

Step 7: We have completed our Spirograph drawing! Try different sized circles, points around the circle, colors, and points of connections to create different types of patterns and have fun! 🙂

Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

For more Math + Art, check out this post on Perspective Drawing here.