Math: Polygons : Quantitative Questions

Bunuel · 2009-11-26T22:26:51-08:00

POLYGONS created by: Bunuel edited by: bb , walker -------------------------------------------------------- https://gmatclub.com/forum/download/file.php?id=10382 This post is a part of GMAT MATH BOOK ] https://gmatclub.com/forum/download/file2.php?id=23736 -------------------------------------------------------- Get The Official GMAT Club's App - GMAT TOOLKIT 2 . The only app you need to get 700+ score! iOS App ] Android App ] -------------------------------------------------------- Types of Polygon Regular A polygon with all sides and interior angles the same. Regular polygons are always convex. Convex All interior angles less than 180°, and all vertices 'point outwards' away from the interior. The opposite of concave. Regular polygons are always convex. Definitions, Properties and Tips • Sum of Interior Angles 180(n-2) where n is the number of sides • For a regular polygon, the total described above is spread evenly among all the interior angles, since they all have the same values. So for example the interior angles of a pentagon always add up to 540°, so in a regular pentagon (5 sides), each one is one fifth of that, or 108°. Or, as a formula, each interior angle of a regular polygon is given by: \frac{180(n-2)}{n} , where n is the number of sides. • The apothem of a polygon is a line from the center to the midpoint of a side. This is also the inradius - the radius of the incircle. polyincircle.gif • The radius of a regular polygon is a line from the center to any vertex. It is also the radius of the circumcircle of the polygon. polycircumcircle.gif GMAT is dealing mainly with the following polygons: Quadrilateral A polygon with four 'sides' or edges and four vertices or corners. 661px-Quadrilaterals.svg.png Types of quadrilateral : Square All sides equal, all angles 90°. See Definition of a square. Rectangle Opposite sides equal, all angles 90°. See Definition of a rectangle. Parallelogram Opposite sides parallel. See Definition of a parallelogram. Trapezoid Two sides parallel. See Definition of a trapezoid. Rhombus Opposite sides parallel and equal. See Definition of a rhombus. Parallelogram A quadrilateral with two pairs of parallel sides. 255px-Parallelogram.svg.png Properties and Tips • Opposite sides of a parallelogram are equal in length. • Opposite angles of a parallelogram are equal in measure. • Opposite sides of a parallelogram will never intersect. • The diagonals of a parallelogram bisect each other. • Consecutive angles are supplementary, add to 180°. • The area , A , of a parallelogram is A = bh , where b is the base of the parallelogram and h is its height. • The area of a parallelogram is twice the area of a triangle created by one of its diagonals. A parallelogram is a quadrilateral with opposite sides parallel and congruent. It is the "parent" of some other quadrilaterals, which are obtained by adding restrictions of various kinds: • A rectangle is a parallelogram but with all angles fixed at 90° • A rhombus is a parallelogram but with all sides equal in length • A square is a parallelogram but with all sides equal in length and all angles fixed at 90° Rectangle A 4-sided polygon where all interior angles are 90° 250px-Rectangle_.png Properties and Tips • Opposite sides are parallel and congruent • The diagonals bisect each other • The diagonals are congruent • A square is a special case of a rectangle where all four sides are the same length. • It is also a special case of a parallelogram but with extra limitation that the angles are fixed at 90°. • The two diagonals are congruent (same length). • Each diagonal bisects the other. In other words, the point where the diagonals intersect (cross), divides each diagonal into two equal parts. • Each diagonal divides the rectangle into two congruent right triangles. Because the triangles are congruent, they have the same area, and each triangle has half the area of the rectangle. • Diagonal=\sqrt{w^2+h^2} where: w is the width of the rectangle, h is the height of the rectangle. • The area of a rectangle is given by the formula Width*Height . A rectangle can be thought about in other ways: • A square is a special case of a rectangle where all four sides are the same length. Adjust the rectangle above to create a square. • It is also a special case of a parallelogram but with extra limitation that the angles are fixed at 90°. Squares A 4-sided regular polygon with all sides equal and all internal angles 90° 220px-Square_(geometry).svg.png Properties and Tips • If the diagonals of a rhombus are equal, then that rhombus must be a square. The diagonals of a square are (about 1.414) times the length of a side of the square. • A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or a parallelogram with equal diagonals that bisect the angles. • If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. (Rectangle (four equal angles) + Rhombus (four equal sides) = Square) • If a circle is circumscribed around a square, the area of the circle is \frac{\pi}{2} (about 1.57) times the area of the square. • If a circle is inscribed in the square, the area of the circle is \frac{\pi}{4} (about 0.79) times the area of the square. • A square has a larger area than any other quadrilateral with the same perimeter. • Like most quadrilaterals, the area is the length of one side times the perpendicular height. So in a square this is simply: area=s^2 , where s is the length of one side. • The "diagonals" method. If you know the lengths of the diagonals, the area is half the product of the diagonals. Since both diagonals are congruent (same length), this simplifies to: area=\frac{d^2}{2} , where d is the length of either diagonal • Each diagonal of a square is the perpendicular bisector of the other. That is, each cuts the other into two equal parts, and they cross and right angles (90°). • The length of each diagonal is s\sqrt{2} where s is the length of any one side. A square is both a rhombus (equal sides) and a rectangle (equal angles) and therefore has all the properties of both these shapes, namely: The diagonals of a square bisect each other. • The diagonals of a square bisect its angles. • The diagonals of a square are perpendicular. • Opposite sides of a square are both parallel and equal. • All four angles of a square are equal. (Each is 360/4 = 90 degrees, so every angle of a square is a right angle.) • The diagonals of a square are equal. A square can be thought of as a special case of other quadrilaterals, for example • a rectangle but with adjacent sides equal • a parallelogram but with adjacent sides equal and the angles all 90° • a rhombus but with angles all 90° Rhombus A quadrilateral with all four sides equal in length. 280px-Rhombus.svg.png Properties and Tips • A rhombus is actually just a special type of parallelogram. Recall that in a parallelogram each pair of opposite sides are equal in length. With a rhombus, all four sides are the same length. It therefore has all the properties of a parallelogram. • The diagonals of a rhombus always bisect each other at 90°. • There are several ways to find the area of a rhombus . The most common is: base*altitude . • The "diagonals" method. Another simple formula for the area of a rhombus when you know the lengths of the diagonals. The area is half the product of the diagonals. As a formula: \frac{d1*d2}{2} , where d1 is the length of a diagonal d2 is the length of the other diagonal. Trapezoid A quadrilateral which has at least one pair of parallel sides. 292px-Trapezoid.svg.png Properties and Tips • Base - One of the parallel sides. Every trapezoid has two bases. • Leg - The non-parallel sides are legs. Every trapezoid has two legs. • Altitude - The altitude of a trapezoid is the perpendicular distance from one base to the other. (One base may need to be extended). • If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same. • If the legs are parallel, it now has two pairs of parallel sides, and is a parallelogram. • Median - The median of a trapezoid is a line joining the midpoints of the two legs. • The median line is always parallel to the bases. • The length of the median is the average length of the bases, or using the formula: \frac{AB+DC}{2} • The median line is halfway between the bases. • The median divides the trapezoid into two smaller trapezoids each with half the altitude of the original. • Area - The usual way to calculate the area is the average base length times altitude. The area of a trapezoid is given by the formula h*\frac{a+b}{2} where a , b are the lengths of the two bases h is the altitude of the trapezoid • The area of a trapezoid is the altitude*median . https://gmatclub.com/forum/download/file2.php?id=23736 -------------------------------------------------------- Get The Official GMAT Club's App - GMAT TOOLKIT 2 . The only app you need to get 700+ score! iOS App ] Android App ] --------------------------------------------------------

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30 Aug 2010, 14:23

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excellent ! - thank you +1

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05 Oct 2010, 11:59

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Thanks Bunuel for such comprehensive notes....

mission2009

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07 Oct 2010, 20:46

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Hi Bunuel,
Nice post can you extend this by including solid figures also.

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08 Oct 2010, 00:33

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mission2009

Hi Bunuel,
Nice post can you extend this by including solid figures also.

math-3-d-geometries-102044.html

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27 Nov 2010, 06:45

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One more masterpiece from Bunuel...... Kudos...!!

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02 Feb 2011, 00:32

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Pkit

Bunuel,

you may add this to your post.

How many diagonals has a polygon with 25 sides?

the formula is:
Number of diagonals in a given polygon is =\(\frac{n*(n-3)}{2}\), where n – is the number of sides.

simple example in a square number of diagonals is =\(\frac{4*(4-3)}{2}=2\)

The below is the concept used to derive the above formula for Number of diagonals .

As we know, to make a diagonal (a line), we need 2 (a pair of) points.
if i have n sided plygon, i need to select different par of points.

This can be doen in \(nc2\) ways
but thease pairs include all the sides also. hence subtract the number of sides (\(n\)), whiich are not diagonals, from the above
==> \(The Number of diagonals\) is
==> \(nc2-n\)
==> \(\frac{n(n-1)}{2} - n\)
==> \(\frac{(n^2-3n)}{2}\)
==> \(\frac{n(n-3)}{2}\)

Regards,
Murali.

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08 Feb 2011, 01:14

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RenukaD

Great post , as always!, by Bunuel. Thanks

Great post !!! :clap:

THANK YOU
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27 Jun 2011, 09:22

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Hi, Super awesome post by the genius again. :thanks

One small doubt. if a DS question talks about a parallelogram and then goes on to say that the diagonals are equal. Does it imply we have a rectangle ?

Further the post says that should the diagonals of a parallelogram be equal and the angles be bisected by it, its a square. But I think this will happen in case of both a rect and a square ?

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27 Jun 2011, 09:24

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Hi, Super awesome post by the genius again. :thanks

One small doubt. if a DS question talks about a parallelogram and then goes on to say that the diagonals are equal. Does it imply we have a rectangle ?

Further the post says that should the diagonals of a parallelogram be equal and the angles be bisected by it, its a square. But I think this will happen in case of both a rect and a square ?

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02 Jul 2011, 05:14

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many thanks for this !

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16 Jul 2011, 00:22

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Great post....Thanks

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17 Jul 2011, 05:22

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great post..thanks Bunuel

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another awesome post... thanks Benuel

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10 Nov 2011, 01:23

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Many thanks!! could find the info only there!! highly appreciate :-D

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03 Dec 2011, 13:25

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Amazing post, thank you!

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04 Jan 2012, 00:28

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Bunuel, thanks once again for your great work.

Although this has already been asked in some form on this thread, I would like to ask you to add a set of sufficient properties that would allow a GMAT taker to determine the type of a quad. I find that this aspect is very moot in the Manhattan Geometry guide. The set of properties should be as minimal as possible in order to be useful on DS questions. For example (please correct me if those properties are not true),

Parallelogram
Given a quad, if its opposite sides are parallel then it is a parallelogram.
Given a quad, it its opposite sides are equal then it is a parallelogram.

Rhombus
Given a quad, if its diagonals are perpendicular bisectors then it is a rhombus.

Square
Given a quad, if its sides are equal then it is a square.
Given a quad, if its angles are equal (90 degrees) then it is a square.
Given a quad, if its diagonals are equal then it is a square.

Thanks in advance.

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28 Oct 2012, 07:14

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Regular polygon's area can be calculated,other polygon's area can also be calculated using graph coordinates,but i have heard that polygon's area can be calculated using sides and external angles is it true?if it is what is the method?

Math: Polygons

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