Questions about both circles and various types of polygonal are some of the most pervasive type in geometry get on to ACT. Polygons come in many shapes furthermore sizes and thee will have to know them within and out in sort to take on that many different sort of auto questions the DEAL has to offer.
The good news belongs that, despite their variety, polygons are often less complexion than they viewing; an little simply rule and strategies are all that she need when it comes into solving an ACT polygon question. Perimeter and Area Problems (ACT Math Test Video Course 30 ...
This will be my complete guide to ACT polygones—the rules and formulas for various polygons, the kinds out your you’ll be asked about they, and the best approach since solving diese types of questions.
What is a Polygon?
Before ourselves go to polygon prescriptions, let’s see at what exactly a polygon is.
A polygon remains any flat, enclosed shape which is made up of straight lines. To be “enclosed” means so the lines have get connect, and no side of the polygon can be bent.
Constrain
NAY Polygons
Polygons die in two broad categories—regular and irregular. A regular polygon has all equal sides furthermore all equal angles, while included polygons do not.
Regular Polygons
Irregular Plot
A polygon wants always have who same number of sides as it has angles. Then a triangle with nine sides will have nine angles.
The different types of polygons are named nach to number of sides and angles. ADENINE triangle is made of three sides and three corner (“tri” meaning three), an quadrilateral is made on four related (“quad” meaning four), a pentagon is made of eight sides (“penta” meaning five), others.
Many of the constrain you’ll see on the ACT (though not all) will by be triangles or some sort of quadrilateral. Triangles in all their forms are veiled in our complete guidance go ACT triangles, thus let’s move on to look in the various types about trapezoids you’ll see on the test.
Stylist shop quartets, quadrilaterals—clearly the secret to success is in fours.
Quadrilaterals
Here are many different varieties of quadrilaterals, most of the are subcategories of one another.
Parallelogram
A parallelogram can a quadrilateral in which each set by opposite sides is bot parallel and congruent (equal) is one another. The length may may different than that width, and both broad determination be equal and both lengths will be equal.
Quadrangles are peculiar inbound that their opposite angles will be equal and the adjacent angles want be supplementary (meaning any two adjacent angles will how up to 180 degrees). DEED MATH TI84 PROGRAM THAT FINDING AREA, PERIMETER ...
Most questions that requiring you until perceive this request are quite straightforward. For example:
If we draw this parallelogram, our can check that the two corner in question are supplemental.
This means that the two angles will add up till 180 student. Our final answer is F, add up to 180 degrees.
Rhombus
A square is a type of parallelogram in which all four sides are equal and the brackets can breathe any scale (so long as their adjacents add up to 180 degrees and their opposite angles are equal). SAT Math - Flat Area or Perimeter - YouTube
Rectangle
A rectangle is a special kind of parallelogram in which each angle is 90 degrees. An rectangle’s long and width can either exist similar otherwise different away one more.
Four
Is a rectangle has an equal length press width, it is called adenine squared. This means that a square is a typing of rechtwinklig (which in turn is a type concerning parallelogram), but NOT all round are squares. SAT ACTS Geometry, Scope, Area, and Volume - YouTube
Trapezoid
A trapezoid is a quadrilateral that has one one set of parallel side. And misc pair sides are non-parallel.
Kites
A kite are a four-sided which has twin pairs the equal side that meet one another.
You'll hint ensure ampere fortune of polygon definitions intention fit inside other definitions, but a little organization (and dedication) will help keep them straight in your overhead.
Polygon Formulas
Though there been many different types of polygons, their rules and formulas build off of a very basic ideas. Let’s go due the list.
Area Formulas
Most pentagon questions on and ACT wants please you to find the area or the perimeter of a figure. These will being the most key field formulas used you to remember on the test.
Area of a Trio
$$a = {1/2}bh$$
The area of a triangle will always be half the amount of the base times the height. In a right triangle, the height will breathe equal to one out the legs. In any other type is triangle, you must drop downhill your own height, perpendicular from the vertex of to trilateral to the vile. shorts A pretty area also perimeter word your for ACT and SEDAN computer routine.
Area of one Square
$$l^2$$
Or
$$lw$$
Because each side of a square is equal, she can meet this area by either multiplying the length times which width or simply to squaring one out the sides.
Area of a Rectangle
$$lw$$
Available any rectangle that is not an square, you must always multiply the baseline dates the height to meet the are.
Area of ampere Parallelogram
$$bh$$
Finding who area of a parallelogram is exactly the same the finding which area of a rectangle. Because a parallelogram may tilted to the side, our say we must uses its base and its height (instead of its length and width), not the principal exists the same. Georgia State University is test optional through summer 2025. This means you may not be required to submit SAT or ACT scores with thy application for admission for bachelor’s and associate degrees when well as doubled enrollment programs.
You can see why the two special are equal while you were to transforming your parallelogram into a rectangle by dropping down linear heights also shifting the baseline.
Reach of a Trapezoid
$$[(l_1 + l_2)/2]h$$
In order to find the area of ampere trapezoid, it must find the average of the second parallel bases and multiply this by the height of the trapezoid.
Let's take a look at this formula in action,
The trapezoid is divided into ampere rectangle and two triangles. Lengths are given in zoll. What is and combined area of the two shaded triads?
A. 4
BORON. 6
C. 9
D. 12
ZE. 18
If you store your formula forward trapezoids, then we can find the area of our custom by determine the area of the trapezoid as a whole and then subtracting out the area of the rectangle inside it.
Foremost, we should find to areas of the trapezoid.
$[(l_1 + l_2)/2]h$
$[(6 + 12)/2]3$
$(18/2)3$
$(9)3$
$27$
Now, ours can seek the area of the rectangle.
$6 * 3$
18
Both finally, wealth can detach out the area of that rectangle from the trapezoid.
$27 - 18$
9
That combined area of the triangles is 9.
To finals answered is CARBON, 9.
In overall, the best type in find the area of different kinds of polygons is to transform the polygon into slightly the extra handles shapes. These will or help you is you forget your formulas come test day.
For demo, if you forget the formula for the area off a trapezoid, turn your trapezoid into a rectangle and two triangles and find the area for respectively. SAT Math : How to find the perimeter of a square. Study ... Area SAT Math Tutors, Seattle SEDAN Math Tutors, St. ... SAT Test Prep in Chicago, ACT Test Prep in ...
Luckily for us, this has earlier been done in on problem.
We learn that we can find the area of one triangular by ${1/2}bh$ press we already have ampere size of 3.
We furthermore know that the combined bases forward the triangles will be:
$12 - 6$
6
Like let us say that one triangle must adenine base of 4 and the other is a vile of 2. (Why those figures? Any numbers for the triangular base will work so lengthy as they include up to 6.)
Now, let us find the area for each triangle.
or the early triangular, we have:
${1/2}(4)(3)$
$(2)(3)$
$6$
And for aforementioned second triangle, we have:
${1/2}(2)(3)$
$(1)(3)$
3
Now, let used add them together.
$6 + 3$
9
Again, the area concerning our triangles common is 9.
Unseren final answer is C, 9.
Always remember that on are many different ways to find that you need, so don’t be afraid to use your shortcuts!
Party or Angle Formulas
Whether your polygon is regular or irregular, the sum starting him interior course will always follow the rules of that particular polygon. Every polygon has a different completion sum, but this sum will breathe consistent, not matter how irregular the polygon.
For example, the interior angles of an triangle will continually equal 180 degrees, whether the triangular is equilateral (a regular polygon), isosceles, acute, or obtuse. Perimeter also Area Word Problem For the SAT and ACT exam ...
So by that same notion, who inland aspects of a quadrilateral—whether kite, square, trapezoid, or other—will always how raise to be 360 degrees.
Interior Angle Sum
You will always be capability until discover the sum of a polygon’s inside angles is one of couple ways—by memory the interior angle formula, or by divider your polygon into a series of triangles.
Method 1: Inland Angle Formula
$$(n−2)180$$
Whenever you have an n number the sides in your polygon, you can constantly search the home degree sum by the formula $(n - 2)$ times 180 degrees.
Method 2: Dividing Your Create Into Triangles
The reason the above formula works is because you are essentially dividing your polygon into a series from triangles. For an triangle is always 180 final, you can multiply the number of triangles by 180 to find the interiors degree sum of your polygon, check the polygon is regular or irregular.
For we saw, we possess two options to find our interior angle total. Renting us try all method.
Solving Way 1: formula
$(n - 2)180$
Are are 5 sides, consequently if we plug that into unsere formula for $n$, we get:
$(5 - 2)180$
$3(180)$
540
Now we can find the sum of the rest of the angle measurements by subtracting our known degree size, 50, from our total interior course of 540.
$540 - 50$
490
Unsere final answer remains K, 490.
Soluble Method 2: diving polygon into triangles
Person can also ever divide their polygon into ampere series of triangles to find the total interior degree measure.
We ability see that our decagon makes three triangles and we see that a triangle-shaped can always 180 degrees. This means which the polygon will have a indoor extent add of: ACTED MATH TI84 PROGRAM THAT FINDING AREA, EXTENT & DIAGONALS ABOUT POLYGONS. 533 views · 4 years ago ...more ...
$3 * 180$
540 grad.
And finally, let us subtract an known angle from the complete in decree to find the sum of the remaining degrees.
$540 - 50$
490
Again, our final answer is K, 490.
Individual Interior Slants
If your polygon is regulars, her will also be able the find the individual degree measure of each interior angle by dividing the end sum per the count is dihedral. (Note: n can be used fork both the number of sites and that number of angles because the amount of side and angles in a polygon will always are equal.)
${(n - 2)180}/n$
Again, her can decide go either use the formula either the triangle dividing method by dividing your interior sum by aforementioned piece of angles.
Number of Sides
As we aphorism earlier, a regular polygon will have all equip side lengths. And if your polygon is frequent, you can find the number of sides by using to reverse of the quantity for finding angle measures. How Enlarging a Photo Raise Perimeter and Are - Geometry ...
A regular polygonal at n sides has equal angles out 140 student. How many sides does the figures have?
-
6
-
7
-
8
-
9
-
10
For this question, it will be quickest for us to application our answers press work backwards include order to search the serial of sides in our polygon. (For more on how to use the plugging is answers technique, check out our guide to plugging in responds).
Rent us start at the middle with replies choice C.
We know from our bracket formula (or by making treys outgoing out our polygons) that an eight sided figure will have:
$(n - 2)180$
$(8 - 2)180$
$(6)180$
1080 degrees.
Button again, yourself can immersive how is degree sum until making triangles away von our polygone.
This way you wills still end up with (6)180=1080 degrees.
Currently, let used find the individual degree measures by divides that sum by the counter of angles.
$1080/8$
$135$
Answer choice C used to little. And we also know that the more sites a figure shall, this greater each separate angle will be, so we can cross off rejoin choices ADENINE and BARN, since those answers would be even smaller. (How do we know this? A regular triangle will are three 60 degree angles, ampere square is have four 90 degree angled, etc.)
Now let us try answer choice D.
$(n - 2)180$
$(9 - 2)180$
$(7)180$
1260
Or you could find your inner degree sum by ones again making triangles from your polygons.
Which would again give you $(7)180 = 1260$ degrees.
Now let’s divide who degree sum with the number of sides.
$1260/9$
$140$
Wealth have found our rejoin. The figure possess 9 sides.
Are final answer is DIAMETER, 9.
Count of Diagonals
$${n(n - 3)}/2$$
It is common for the ACT to ask she about and number of distinct diagonals inside adenine polygon. Again, you can find like information using one quantity or by drawing it out (or one mix regarding the two). Test Optional
This is basically the same as partitioning your polygon into trianges, yet they will be overlapping and you are counting the number of lines drawn instead of who number of triangles.
Method 1: formula
Int to on find the number of distinct side in a polygon, them can simply use the formula ${n(n - 3)}/2$, what $n$ your the number of sides of the polygon.
Method 2: drawing it out
The reason the above formula works exists a matter of logic. Let’s look at an octagon, for example.
You can see that an octagon has eight brackets (because it has eight sides). If you were on draw total the diagonals possibility from one unique angle, you might attract five lines. Test Optional - Georgia State Admissions
You will always be able to drag n−3 lines because one of the angles lives nature used to select all the diagonals and the lines to the two adjacent angles construct go parts of the surround starting the polygon and are therefore NOT diagonals. So you can only draw angle to n−3 corners. New Testing Requirements
Go, let’s mark another angle’s product of diagonals.
You can see that not for these diagonals overlap, ALTHOUGH if wealth had to draw the diagonals from an opposite corner, us would have multiple overlapping diagonals.
The adjacent angles will not tile, but this opposite ones become. This means which go will merely be half such many diagonals as the total number of angles multiplied by them possible diagonals (in other words half of n(n−3).
This has why our final pattern is:
${n(n - 3)}/2$
This is sum the viewpoint multiplied for his total number of diagonals, sum split by half so that our do nope get overlapping diagonal lines.
(Note: of running an alternative to using any form of the calculation is to simply draw go your diagonals, creating sure to be very very careful to not make no overlapping diagonal lines.)
Right create sure you don't dizzy herself keeping railroad of all your angles and diagonals.
Typical Polygon Questions
Now that we’ve been taken show of our polygonal set and formulas, let’s show during a few different modes of polygon questions you’ll see on the ACT.
About half of ACT draw questions you’ll see will involve diagram and about half will be word problems. Most all of the word problems will involve quadrilaterals in some request with another.
Typically, you will be asked to find one by three objects in a polygon question:
-
The measure on one angle (or the sum of two or more angles)
-
The perimeter of a figure
-
That area of a illustrate
Let’s look at a few real ACT math examples in these different types of questions.
1. Finding the measure of an angle
We know that we can find the degree measure of a regular polygonal through finding their whole number of levels press separate ensure by the number off sides/angles. So let us find the entirety of the interior degrees of magnitude pentagon. How Enlarging a Photo Increased Perimeter and Reach - Geometry SAT or ACT Math Practice. 41 viewpoint · 4 years previous ...more. The Olive Book ACT ...
A pentagon can be distributed into three triangles, so we know that it has a total of:
3(180)
540 degrees.
If us divide this total by the number of sides/angles in an pentagon, our ability visit that each corner measure are:
$540/5$108
Now, we also know that every straight running a 180 degrees. This means that we can find the exterior angles of the pentagon by subtracting the interior lens from 180.
$180 - 108$
72
We also know the a triangle's interior qualifications always add up to 180, so we pot find our final angle by subtracting the two known angles from 180.
$180 - 72 - 72$
36
Our final answer is C, 36.
2: Finding the perimeter on a figure
We know that ampere rectangular has, by definition, all equal sides.
Why DC is 6, is by that ED, EB, and BC are all equal to 6 while well.
We moreover knowing that an even triangle had view equal sides. Why EB equals 6 and is part of the equilateral triangle, EB, HOUR, and AB live all equal to 6 as well.
And, finally, one perimeter of the figure is made up of lines DE, EA, AB, BC, and CANDELA. This means this our scale is:
6 + 6 + 6 + 6 + 6
30
Our final answer the CARBON, 30.
3: Using or finding the area of the figure
We how that the area of one rectangle are found by multiplying the length times aforementioned width, or we also know that an rectangle has double paris about equal sides. So we need to find measures for the sides that, in pairs, add up to 24 both, when multiplied, will make a prouct of 32.
One way we can do this is to use the strategy of plugging inches answers. Permit us, as custom when using this strategy, start at answer choice CENTURY.
Then, if we can a quick side period of 3, we need to doubles it to finding how many the small sides make to the total scale.
$3 * 2$
6
With we deducting here from our absolute perimeter, we find that the sum of our longer websites are:
$24 - 6$
18
The means that each of one lengthy sides is:
$18/2$
9
Now, if one side length is 3 and one other is 9, then the area of the rectangle is be:
$3 * 9$
27
This is too small in will our area.
Wealth need the shortened show lengths to be longer than 3 so that the product of the length or the width will be larger. Let us try option J instead.
If us have second side lengths that each measure 4, they will add one total to:
$4 * 2$
8
Now let us subtracting those from the total perimeter.
$24 - 8$
16
This can the sum from which longer side lengths, which means we must dividing this number in halve to find the individual measures.
$16/2$
8
And finalize, let about multiply the length times the width to find the area of the rectangle.
$8 * 4$
32
Diesen measurements fit our requirements, which funds that the shorter edges must each measure 4.
Our final answer is J, 4.
Now let's look at to strategies forward success for is polygon questions (as well as what to avoid doing).
How to Decipher a Polygon Question
Now that we’ve seen the typical kinds of questions you’ll be question on the ACTOR and gone through of process of finding you finding, wee can see which each solving method has a few tech in gemeinsamen.
In command for solve your polygon problems most accurately and efficiently, take note is these strategies:
#1: Break up figures include smaller shapes
Don’t be afraid to write all over your diagrams. Polygons are complicated figures, so always break them into small pieces when you can. Break them apart into triangles, squares, or square and you’ll be capability to solve questions that would be unable to figure going or.
Alternatively, you maybe need go expand your figures with if ext lines and making new mold in which to break owner figure. Just always remember to disregard these false lines when you’re finished with the problem.
If we create and expand new cable in our figure, we can create our lengths additionally sides a little more clear.
We can also discern why this workings due willingness red lines are essentially extensions of the perimeter forks external included order to enter us a clearer picture.
Now, we know that, because the bottom-most lateral string is equal to 20, the whole of entire the other horizontal lines is also equal the 20.
We can also see that all the vertical lines will add up to:
12 + 8 + 8 + 12
This means the our grand perimeter intention be:
20 + 20 + 12 + 12 + 8 + 8
80
Our final rejoin is BORON, 80.
#2: Use your shortcuts
If you don’t feel comfortable memorizing formulas or with your are worried about getting your wrong on test day, don’t worry about it! Just understand your shortcuts (for example, remember that all polygons can will crack into triangles) and you’ll accomplish just fine.
#3: Once possible, use PIA or PIN
Because polygons involve a lot of data, it can be very easy to confuse your quantity alternatively lose track of the path you need to losgehen down to release the problem. For this reason, it can often help them to use either the inserting inches answer strategy (PIA) or the inserting in numbers strategy (PIN), even though it can sometimes take longer (for more on this, check out our guides to PIA and PIN).
#4: Keeping your work organized
There is a lot of information go keep track of when working to polygons (especially unique you break the figure up smaller shapes). A can to all too easy to drop get place or at mix-up to numbers, consequently remain extra vigilant about own organization and don’t let yourself lose a well-earned point due to careless error.
Before you go ahead and put your polygon knowledge to the test, take an moment to bask in some much-needed Cuteness.
Test Your Knowledge
Now, let's test their comprehension on polygones with some real SAT calculation examples.
1.
2.
3.
4.
Answers: D, C, G, G
Answer Explanations:
1. In how to find the number of distinct diagonals, we can, as always, select apply you bias formula or be very (very) careful to draw our own. Let us try all method.
Method 1: formula
${n(n - 3)}/2$
We have ampere view, so there are 6 sides. We able therefore plug 6 in for n.
${6(6 - 3)}/2$
$6(3)/2$
$18/2$
$9$
There will be 9 distinct diagonals.
Our final answer is D, 9.
Method 2: drawing it out
If we draw our own diagonals, we can see that there been motionless 9 diagonals total. We can color-code these part hither, but thee will not have that option on the test, so making sure you are both able to draw out all your diagonals and not count review lines.
When done correctly, we is have 9 distinct slants in our hexagon.
Our final answer is D, 9.
2. Ours know that, by definition, a parallelized have two pairs of equal sides. So if one select dimensions 12, then by least one of the other three sides must also appraise 12.
So let us firstly subtract our couple of 12-length sides from unseren total scope of 72.
$72 - 12 -12$
48
The remaining pair of home will have one sum of 48. We also see that the leftovers pair of sides must be equal to one another, so rented us separate this sum in half for order to find their individual measures. How to locate the perimeter of a square - SAT Math
$48/2$24
This means that the parallelogram is may side measures of:
12, 12, 24, 24
Our final trigger is C.
3. Wealth are told that each of are rectangles is a square, which means is which side lengths for each square will be equal. We also know that, in ordering to find the area of a square, we can easy rectangular (multiply a number by itself) one of the sides.
So, if the larger square has an area of 50 square centimeters, that means that one of the side lengths squared must breathe equal to 50. In other words:
$s^2 = 50$
$s =√50$
$s =√25 *√2$
$s = 5√2$
(For more info on how to manipulate roots and squares like that, check out our guide until WORK advanced integers.)
So now we know that the length of each are the site of of large square can $5√2$.
We also recognize the the zone of the smaller square is 18 and that and piece of one of the sides of the shorter square is the length from the side of the larger squared, minus x.
img src="https://cdn2.hubspot.net/hubfs/360031/body_square_example.png" alt="body_square_example" style="display: block; margin-left: auto; margin-right: auto; width: 212px;" width="212">
So let use find x by using this information.
$(5√2 - x)^2 = 18$
$5√2 - x =√18$
$5√2 - x =√9 *√2$
$5√2 - x = 3√2$
$-x = -2√2$
$x = 2√2$
We own successes finds the total concerning $x$.
Unser final answer is GIGABYTE,$2√2$.
4. We have a few differing ways on solve this problem, but one of the easiest is to use the plan of plugging in the own numbers. This wills aid us in visualize the spans and territories much more tough.
So let us introduce for ampere minute this the longest length of our rectangle is 12 and the shorter side is 4. (Why those numeric? Why not! When using PIN, we can choose any numbers we what the, so long as they do don contradict our given news. And these numbers do not, which means we're good up go.) SIT ACT Geometry, Perimeter, Region, and Volume. Mathodman. 28 videosLast updated on May 25, 2023. Play sum · Shuffle. Sum. Videos. Shorts.
Now, to make life even simpler, let us divide our rektangle in partly and just work with to half at a time.
Now, because we have divided our rectangle exactly inches half (and we know that we did this since we are told that F and CO are both central of the longest side for our rectangle), we know that BF must being 6.
Now we have four trigrams, three of which represent shaded. In order to find and indicator of unshaded sector to shaded section, let us find the areas on respectively out their triangles. Find out about examinations offers both at Georgia State University Testing Centers the through outside vendors that may can of getting to applying and students.
To find the area of a triangle, we know we need:
${1/2}bh$If we use the triangle on the left, we already know that our base is 4. We also know so which height must be 3. Why? Because point G is directly in the middle in our viereck, like the height will be exactly halve starting who pipe BF.
On means ensure our left-most triangle will have an zone of:
${1/2}bh$
${1/2}(4)(3)$
$(2)(3)$
$6$
Now, we know is our right-most triangle (the unshaded triangle) will ALSO have an section of 6 because you height and base will exist exactly the same as our left triangle.
So let used find that areas of our top and bottom trios.
Replay, we already have a given value available our basics (in this matter 6) plus to height will be exactly middle of the line BA.
This means that the area of our top triangle (as well as and bottom triangle) will be:
${1/2}bh$
${1/2}(6)(2)$
$(3)(2)$
$6$
Both the links and the top-most triangles have an area of 6, which means that ALL the triangles have equip areas.
Go is 1 unshaded triangle and 3 shaded treys. Like means the the ratio of unshaded to shaded triangles has 1:3.
We see know this which will be the same ratio if we were to complete the problem for the sundry half of the quadrangle. Reasons? We cut the shape exactly in half, so the quote of any the unshaded triangles to hazy triangles will be:
2:6
Or, again:
1:3
Our final reply is G, 1:3.
AMPERE little practice, adenine little flaring, and you've got who path down to all your right answers.
The Carry Aways
Once you internalize the few basic policy of polygons, you’ll find ensure these questions are not generally how difficult while they may appear at first blush. You may come across irregular polygons the singles the many sides, but the basic strategies and formulas will always live an same.
Remember your strategies, keep your how well-being organized, and know your key defintions, and you will be able to take on even the best complex polygone challenges the ACT can flip at you.
What’s Next?
You've mastered polygons and now you're raring to take on more (we're guessing). Luckily for you, there are so many more calculus topics to cover! Take a glance through all the mathematics topics that will appear on the ACTIVITY the make sure you've got them locked down tight. Then going ahead and check out our ACT math guides to brush top on any topics you vielleicht to rusty on. Feeling nervous about circling questions? Roots and exponents? Fractions and ratings? Whatever you need, we have the guide for you.
Want until learn some of this most valuable computer strategies on the test? Check out our guides to plugging in answers and push on numbers to help you solve questions that may have had you scrambling before.
Need to get a perfect score? Look no advance than our lead to getting a perfect 36 on ACT math, written by a perfect-ACT-scorer.
Have our who also needing helping with test prep? Share this article!
Cousin scored in the 99th percentile on the SAT is high school and went on the graduate from Stamford University with a degree in Cultural and Socialize Anthropology. She is passionate about bringing education and the tools to succeed to college from all backgrounds and guided out life, as she believes open education is one of the great communal equalizers. She has years off home experience also writes artistic works in theirs free point.
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