Dilations: Scale Factor & Points Other than Origin

Hi there and welcome to MathSux! Today we are going to break down dilations; what they are, how to find the scale factor, and how to dilate about a point other than the origin. Dilations are a type of transformation that are a bit different when compared to other types of transformations out there (translations, rotations, reflections). Once a shape is dilated, the length, area, and perimeter of the shape change, keep on reading to see how! And if you’re looking for more transformations, check out these posts on reflections and rotations. Thanks so much for stopping by and happy calculating! 🙂

What are Dilations?

Dilations are a type of transformation in geometry where we take a point, line, or shape and make it bigger or smaller, depending on the Scale Factor.

We always multiply the value of the scale factor by the original shape’s length or coordinate point(s) to get the dilated image of the shape. A scale factor greater than one makes a shape bigger, and a scale factor less than one makes a shape smaller. Let’s take a look at how different values of scale factors affect the dilation below:

Scale Factor >1 Bigger

Scale Factor <1 Smaller

Scale Factor=2

 In the below diagram the original triangle ABC gets dilated by a scale factor of 2.  Notice that the triangle gets bigger, and that each length of the original triangle is multiplied by 2.

Dilations

Scale Factor=1/2

Here, the original triangle ABC gets dilated by a scale factor of 1/2.  Notice that the triangle gets smaller, and that each length of the original triangle is multiplied by 1/2 (or divided by 2).

Dilations

Properties of Dilations:

There are few things that happen when a shape and/or line undergoes a dilation.  Let’s take a look at each property of a dilation below:

1. Angle values remain the same.

2. Parallel and perpendicular lines remain the same.

3. Length, area, and perimeter do not remain the same.

Now that we a bit more familiar with how dilations work, let’s look at some examples on the coordinate plane:

Example #1: Finding the Scale Factor

Step 1: First, let’s look at two corresponding sides of our triangle and measure their length.

Dilations

Step 2: Now, let’s look at the difference between the two lengths and ask ourselves, how did we go from 3 units to 1 unit?

Remember, we are always multiplying the scale factor by the original length values in order to dilate an image. Therefore, we know we must have multiplied the original length by 1/3 to get the new length of 1.

Dilations

When it comes to dilations, there are different types of questions we may be faced with.  In the last question, the triangle dilated was done so about the origin, but this won’t always be the case.  Let’s see how to dilate a point about a point other than the origin with this next example.

Example #2: Dilating about a Point other than the Origin

Dilations

Step 1: First, let’s look at our point of dilation, notice it is not at the origin! In this question, we are dilating about point m!  To understand where our triangle is in relation to point m, let’s draw a new x and y axes originating from this point in blue below.

Dilations

Step 2: Now, let’s look at coordinate point K, in relation to our new axes.

Step 3: Let’s use the scale factor of 2 and the transformation rule for dilation, to find the value of its new coordinate point. Remember, in order to perform a dilation, we multiply each coordinate point by the scale factor.

Step 4: Finally, let’s graph the dilated image of coordinate point K. Remember we are graphing the point (6,4) in relation to the x and y-axis that stems from point m.

Dilations

Check out these dilation questions below!

Practice Questions:

1) Plot the image of Point Z under a dilation about the origin by a scale factor of 2.

2) Triangle DEF is the image of triangle ABC after a dilation about the origin. What is the scale factor of the dilation?

Dilations

3) Point L is dilated by a scale factor of 2 about point r. Draw the dilated image of point L.

Dilations

4) Line DE is the dilated image of line AB. What is the scale factor of the dilation?

Solutions:

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

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Looking for more Transformations? Check out the related posts below!

Translations

Rotations

Reflections

Algebra 2 Cheat Sheet & Review

Ahoy and welcome math friends! For the latest installment, here is the Algebra 2 Cheat Sheet & Review made just for you to prepare for finals. On this page, you’ll also find links to the come math friends! For the latest installment, here is the Algebra 2 lesson playlist, the NYS Algebra 2 Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! 🙂

Algebra 2 Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to graphs, it’s on here.

Algebra 2 Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Algebra 2 below. It includes every MathSux video related to Algebra 2 and will be sure to help you ace the test!

Algebra 2 Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Algebra 2 Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra 2 library organized by topic here.

Algebra 2 Cheat Sheet & Review


Still got questions? No problem! Don’t hesitate to comment with any questions below. Also, if you find you need some motivation, check out my 6 tips and tricks for studying math here! Thanks for stopping by and happy calculating! 🙂

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Get everything you need to know with this Algebra 2 Cheat Sheet and Review! Download and print the pdf for reviewing Algebra 2 or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Geometry Cheat Sheet & Review

Greeting math peeps! As promised here is the Geometry Cheat Sheet and Review made just for you to prepare for finals. On this page, you’ll also find links to the Geometry lesson playlist, the NYS Geometry Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! 🙂

Geometry Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to shapes, it’s on here.

Geometry Review Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Geometry below. It includes every MathSux video related to Geometry and will be sure to help you ace the test!

Geometry Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Geometry Math Lessons for Review:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Geometry library organized by topic here.

Geometry Cheat Sheet & Review

Still got questions? No problem! Don’t hesitate to comment with any questions below. Also, if you find you need some motivation, check out my 6 tips and tricks for studying math here! Thanks for stopping by and happy calculating! 🙂

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Get everything you need to know with this Geometry Cheat Sheet and Review! Download and print the pdf for reviewing Geometry or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Algebra Cheat Sheet & Review

It’s that time of year again, summer is coming, the vacation vibes are calling, but so, unfortunately, are the test cramming and non-stop class reviewing that is coming our way. Nothing like going over topics mentioned at the beginning of the school year to bring us down. How is one supposed to remember everything? Fear not, because I have made a special cheat sheet and review for Algebra, (with Geometry and Algebra 2/Trig. soon to be on the way). I hope you’re staying safe, cool, and calm as the end of the year approaches. Good luck on finals and tests and happy calculating! 🙂

Algebra Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to parabolas, it’s on here.

Algebra Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Algebra below. It includes every MathSux video related to Algebra and will be sure to help you ace the test!

Algebra Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Algebra Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra library organized by topic here.

Algebra Cheat Sheet & Review

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

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Get everything you need to know with this Algebra Cheat Sheet and Review! Download and print the pdf for reviewing Algebra or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Similar Triangles: AA, SSS, & SAS

Happy Wednesday math peeps! In today’s post, we are going to go over Proving Similar Triangles, by going over:

1) What it means when two triangles are similar?

2) How to prove two triangles similar?

3) How to find missing side lengths given triangles are similar?

For even more practice, don’t forget to check out the video and practice problems below. Happy calculating! 🙂

What are Similar Triangles?

When two triangles have congruent angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but as long as they have the same angles and the sides are in proportion, they are similar! We use the “~” to denote similarity.

In the Example below, triangle ABC is similar to triangle DEF:

Similar Triangles

How can we Prove Triangles Similar?

There are three ways to prove similarity between two triangles, let’s take a look at each method below:

Angle-Angle (AA): When two different sized triangles have two angles that are congruent, the triangles are similar.  Notice in the example below, if we have the value of two angles in a triangle, we can always find the third missing value which will also be equal.

Similar Triangles

Side-Side-Side (SSS): When two different sized triangles have three corresponding sides in proportion to each other, the triangles are similar. 

Similar Triangles

Side-Angle- Aside (SAS): When two different sized triangles have two corresponding sides in proportion to each other and a pair of congruent angles between each proportional side, the triangles are similar. 

Similar Triangles

Let’s look at how to apply the above rules with the following Example:

Step 1: Since, we know the triangles ABC and DEF are similar, we know that their corresponding sides must be in proportion! Therefore, we can set up a proportion and find the missing value of length EF by cross multiplying and solving for x.

Similar Triangles

Practice Questions:

1) Are the following triangles similar?  If so, how? Explain.

2) Are the following triangles similar?  If so, how? Explain.

3) Given triangle ABC is similar to triangle DEF, find the side of missing length AB.

4) Given triangle ABC is similar to triangle PQR, find the side of missing length AC.

Solutions:

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

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Check out more posts on Geometry here!

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Construct the Altitude of a Triangle

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

Finite Geometric Series Formula: Algebra 2

Happy Wednesday math friends! In today’s post we are going to go over what finding the sum of a finite geometric series means and then use the finite geometric series formula to solve an example step by step. If you need a refresher on geometric sequences before tackling these types of questions, don’t hesitate to check out this post here. Maybe you’re looking for infinite geometric series, check out the previous link? Also, don’t forget to check out the video that explains everything you see here as well as practice questions below. Happy calculating! 🙂

What does it mean to find the “Sum of the Finite Geometric Sequence”?

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

Example:

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

Example:

More specifically, what if we wanted to find the sum of the first 20 terms of the above Geometric Sequence?  How would we calculate that?  That’s where our Finite Geometric Series Formula will come in handy!

Finite Geometric Series Formula:

Finite Geometric Series Formula

Now that we have a formula to work with, let’s take another look at our question and apply our finite geometric series formula to answer the solution:

Finite Geometric Series Example:

Step 1: First let’s write out our formula and identify what each part represents/what numbers need to be fill in.

Finite Geometric Series Formula

Step 2: Now let’s fill in our formula and solve with the given values.

Finite Geometric Series Formula

Try the similar practice questions below!

Practice Questions:

1) Find the sum of the first 15 terms of the following geometric sequence:

4, 12, 36, 108, ….

2) Find the sum of the first 12 terms of the following sequence and round to the nearest tenth:

128, 64, 32, 16, ….

3) Find the sum of the first 18 terms of the following geometric sequence and round to the nearest tenth:

400, 100, 25, 6.25, ….

4) Find the sum of the first 12 terms of the following geometric sequence:

3, 6, 12, 24, ….

Solutions:

1) 28,697,812

2) 255.9

3) 533.3

4) 12,285

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

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Check out similar posts for Algebra 2/Trig. here! And if you want to learn about even more sequences, check out the link here!

Related Posts:

Finite Arithmetic Series

Infinite Geometric Series

Geometric Sequences

Arithmetic Sequences

Recursive Sequences

How to Construct a 45 Degree Angle with a Compass

Greeting math friends and welcome to another wonderful week of MathSux! In today’s post we are going to break down how to construct a 45 degree angle with a compass. We will take this step by step starting with a simple straight edge, then we will create a 90 degree angle, and finally we will bisect that 90 degree angle to get two 45 degree angles. If you have any questions please don’t hesitate to check out the video and step-by-step GIF below. Thanks so much for stopping by and happy calculating! 🙂

How to Construct a 45 degree Angle with a Compass

How to Construct a 45 Degree Angle with a Compass:

Step 1: Using a straightedge, draw a straight line, labeling each point A and B.

Step 2: Using a compass, place the point of the compass on the edge of point A and draw a circle.

Step 3: Keeping the same length of the compass, take the point of the compass to the point where the circle and line AB intersect. Then swing compass and make a new arc on the circle.

Step 4: Keeping that same length of the compass, go to the new intersection we just made and mark another arc along the circle.

Step 5: Now, take a new length of the compass (any will do), and bring it to one of the intersections we made on the circle.  Then create a new arc above the circle by swinging the compass.

Step 6: Keep the same length of the compass and bring it to the other intersection we made on our circle.  Then create a new arc above the circle.

Step 7: Mark a point where these two lines intersect and using a straight edge, connect this intersection to point A. Notice this forms a 90º angle.

Step 8: Now to bisect our newly made 90º angle, we are going to focus on the pink hi-lighted points where the original circle intersects with line AB and our newly made line.

Step 9: Using a compass (any length), take the compass point to one of these hi-lighted points and make an arc.

Step 10: Keeping that same length of the compass, go to the other hi-lighted point and make another arc.

Step 11: Now with a straight edge, draw a line from point A to the new intersection of arcs we just made.

Step 12: Notice we split or 90º angle in half and now have two equal 45º angles?!

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! 🙂

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

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Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links! And if you’re looking for even more geometry constructions, check out the link here!

Multiplying Radicals Expression

Hi everyone and welcome to MathSux! In today’s post we are going to explore the rules for multiplying radicals, mainly focusing on multiplying binomials expressions that contain radicals. We will go over several types of examples in this post starting with the basics and working our way up to expanding radical binomial expressions and simplifying. If you need a quick review on how to expand binomials using the distributive property/box method, check out this post here. Also, be sure to check out the video and practice questions below. Thanks so much for stopping by and happy calculating! 🙂

What Are the Rules for Multiplying Radicals?

  1. Multiply the coefficients.
  2. Multiply the term or terms under the radical.
  3. Simplify the solution whenever possible.
Multiplying Radicals

Check out more completed Examples of multiplying radicals that involve some simplifying below:

Multiplying Radicals

The above shows a few simple cases of multiplying radicals but let us take a look at how to multiply radicals that are placed within a binomial within these next Examples.

Step 1: To solve this, we are going to use the distributive property to multiply each term by each term, sometimes known as FOIL (for more on the FOIL method, check out this post, here.)

Multiplying Radicals

Step 2: Before we say we are done and have a solution, notice that we can simplify this even more by combining like terms, 12+2=14, we can also combine the like radical terms below because they have the same value of 2 under the radical.

Multiplying Radicals

Now, let’s take a look at another type of Example, where we need to first expand binomials that contain radicals.

Step 1: Notice in with this example, we are going to need to expand this binomial and then use the distributive property (or FOIL) to multiply each term by each term.

Multiplying Radicals

Step 2: In this case, notice that we can simplify our answer even more by combining like terms.

Multiplying Radicals

Try the following practice questions on your own to truly master the topic!

Practice Questions:

Solutions:

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! 🙂

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Focus and Directrix of a Parabola: Algebra 2

Happy Wednesday math friends! In today’s post are going to go over what the focus and directrix of a parabola are and how to find them. We will also take a look at how to find the equation of a parabola when given the focus and directrix. You may already be familiar with parabolas and quadratic equations, but with this post we will dive deeper into the definition of a parabola and understand it on a whole new level! Don’t forget to check out the video and practice questions, below. Thanks for stopping by and happy calculating! 🙂

Parabola? Vertex? Focus? Directrix?  What does it all mean?!

Focus and Directrix of a Parabola:

Parabola: A set of all points in a plane that are equidistant from a given point (the focus) and a given line (the directrix). The equation of a parabola in relation to the focus and the directrix is:

Vertex: The maximum or minimum point on a parabola.  This coordinate point always lands on the parabola itself. Takes the coordinate form (a,(.5(b+k))) from the parabola equation above.

Focus: A coordinate point that is “inside” the parabola that has the same distance from the vertex as it does the distance between the vertex and directrix. Usually denotes as (a,b) within the parabola equation above.

Directrix: A horizontal line denoted as, “y= ” or “k=  “, that is the same distance from the vertex when compared to the distance between the vertex and the focus.

The above definitions explain that every point on a parabola is equidistant from the vertex to the focus as it is to the directrix.  In the picture below, we can see that the vertex is 2 units away from the focus and 2 units away from the directrix. They are equidistant:

Focus and Directrix of a Parabola:

This phenomenon, works not just the vertex, but for each and every point found on a parabola!

We can see below the equidistance between several points on the parabola that compare the distance between the vertex to the directrix and the distance between the vertex to the focus. Notice that they are all the same!

Focus and Directrix of a Parabola:

Ready to test our new knowledge with the following Example?

Step 1: First, let’s sketch out our parabola with the given information.  We know that the vertex is at point (2,-1) a point that lies on the parabola itself, so let us map that out.  We also know that the directrix is k=3 and is represented as a line as y=-3.

Focus and Directrix of a Parabola:

Step 2: Next, let us measure the distance between the directrix and the vertex.  The distance between the vertex and directrix is 2 units. That means we must also measure the 2 units on the opposite side of the vertex to find the value of the focus. This leads us to the point (2,1).

Focus and Directrix of a Parabola:

Now for part two of our question, how to find the equation of a parabola now that we have the value of the focus and the given directrix.  For this, we will need to use the funky looking equation for a parabola mentioned earlier in this post.

Step 1: First we need to gather all of our information, the formula for the equation of a parabola , the given directrix, k=-3 and the focus we found in the previous example (2,1) which corresponds to the formula as a=2 and b=1.

Step 2: Now, let’s plug everything into our formula where a=2, b=1, and k=-3, to find the equation to our parabola:

Practice Questions:

Solutions:

Focus and Directrix of a Parabola:

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

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Check out similar posts for Algebra 2/Trig. here!

Looking for more on Quadratic Equations and Functions? Check out the following Related posts!

Factoring Review

Factor by Grouping

Completing the Square

4 Ways to Factor a Trinomial

Is it a Function?

Imaginary and Complex Numbers

Quadratic Equations with 2 Imaginary Solutions

What is the Discriminant?

How to find the Area of a Parallelogram: Geometry

Hey math peeps! In today’s post, we are going to go over how to find the area of a parallelogram. There is an easy formula to remember, A=bh, but we are going to look at why this formula works in the first place and then solve a few examples. Just a quick warning: The following examples do use special triangles and if you are need of a review, check out the posts here for 45 45 90 and 30 60 90 special triangles. Also, don’t forget to watch the video and try the practice problems below. Thanks so much for stopping by and happy calculating!

Area of a Parallelogram Formula:

How to find the Area of a Parallelogram

Why does the Formula for Area of a Parallelogram work?

Did you notice that the formula for area of a parallelogram above, base times height, is the same as the area formula for a rectangle?  Why?

If we cut off the triangle that naturally forms along the dotted line of our parallelogram, rotated it, and placed it on the other side of our parallelogram, it would naturally fit like a puzzle piece and create a rectangle! Check it out below:

How to find the Area of a Parallelogram

Now that we know where this formula comes from, let’s see it in action in the examples below:

Example #1:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Fill in the formula with values found on our parallelogram, b=12 inches h=4 inches, and multiply them together to get 48 inches squared.

How to find the Area of a Parallelogram

That was a simple example, but lets try a harder one that involves special triangles.

Example #2:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Label the values found on our parallelogram, b=10 ft and notice that we are going to need to find the value of the height.

Step 3: In order to find the value of the height, we need to remember our special triangles! We are not given the value of the height, but we are given some value of the triangle that is formed by the dotted line.  Let us take a closer look and expand this triangle:

Step 4: We can add in the missing 45º degree value so that our triangle now sums to 180º.

Step 5: Remember 45 45 90 special triangles(If you need a review click the link). Because that is exactly what we are going to need to find the value of the height! Below is our triangle on the left, and on the right is the 45 45 90 triangle ratios we need to know to find the value of the height.

Based on the above ratios, we can figure out that the height value is the same value as the base of the triangle, 2.

Step 6: If we place our triangle back into the original parallelogram, we can plug in our value for the height, h=2, into our formula to find the area:

How to find the Area of a Parallelogram

When you’re ready, check out the practice questions below!

Practice Questions:

Find the area of each parallelogram:

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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