Events & Promotions
|
|
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Mar 2709:30 AM PDT
-10:30 AM PDT
Preparing for the GMAT? Avoid these 5 major mistakes that can lower your score and hurt your MBA dreams. In this expert panel discussion, 4 top GMAT coaches - GMAT Ninja, Jeff Miller from TTP, Rida Shafiq from eGMAT, and Chris Gentry from Manhattan Prep- Mar 27
10:00 AM PDT
-11:00 AM PDT
What happens after getting an MBA? Is it worth the investment? To find out, I interviewed 10 GMAT Club professionals who have completed their MBAs and built careers in consulting, finance, tech, and entrepreneurship. Their answers might surprise you! 
Mar 1910:00 AM EDT
-11:59 PM EDT
Get a massive $250 off the TTP OnDemand GMAT course by using the coupon code FLASH25 at checkout. If you prefer learning through engaging video lessons, TTP OnDemand GMAT is exactly what you need.- Mar 26
10:00 AM PDT
-11:00 AM PDT
Confused about how to get the most from your GMAT mock tests? In this video, learn the perfect strategy to analyze GMAT mocks effectively and boost your score. We break down the common mistakes GMAT aspirants make when using mock tests 
Mar 2911:30 AM EDT
-12:30 PM EDT
Learn how Keshav, a Chartered Accountant, scored an impressive 705 on GMAT in just 30 days with GMATWhiz's expert guidance. In this video, he shares preparation tips and strategies that worked for him, including the mock, time management, and more.
Mar 3005:30 AM PDT
-07:30 AM PDT
We know Strengthen and Weaken questions account for more than 50% of the CR questions on the GMAT. With CR becoming even more important on GMAT Focus, it's time you strengthen your weaknesses with an approach that improves your solving time and accuracy!
Mar 3011:00 AM IST
-01:00 PM IST
Attend this session to evaluate your current skill level, learn process skills, and solve through tough Arithmetic questions.
Mar 3110:00 AM EDT
-11:59 PM EDT
The Target Test Prep team has recently introduced TTP OnDemand GMAT, a 400-hour live masterclass video course created by Scott Woodbury-Stewart, founder and CEO of Target Test Prep.
03 Jun 2015, 04:05
Kudos
Bookmarks
INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT
REGULAR POLYGONS AND THE IRREGULAR ONES
By Karishma, Veritas Prep.
Continuing our Geometry journey, let’s discuss polygons today.
Some years back, I used to often get confused in the polygon sum-of-the-interior-angles formula if I had to recall it after a gap of some months because I had seen two variations of it:
Sum of interior angles of a polygon = (n – 2)*180
Sum of interior angles of a polygon = (2n – 4)*90
Now, I don’t want you to judge me. Of course, in the second formula, 2 has been removed from 180 and multiplied to the first factor. It is quite simple so why would anyone get tricked here, you wonder? The problem was that after a few months, I would somehow remember (2n – 4) and 180. So I was mixing up the two and I wasn’t sure of the logic behind this formula. That is until I came across the simple explanation of this formula in our Veritas Prep Geometry book (the one which explains how you can divide every polygon with n sides into (n – 2) triangles and hence get the sum of (n – 2)*180). Now it made perfect sense! I couldn’t believe that I had not come across that explanation before and had just learned up (well, tried to!) the formula blindly. So now I ensure that all my students understand every formula that I teach them.
Usually, we are given a regular polygon and we need to find the measure of interior angles or the number of sides. But what if we are given a polygon instead, not a regular polygon. Does this formula still apply? We wouldn’t know if we didn’t understand how the formula came into being. But since we know that we obtain the formula by dividing the polygon into (n-2) triangles, we know that the sum of all interior angles of a triangle is 180 irrespective of the kind of triangle. So it doesn’t matter whether the polygon is regular or not. The sum of all interior angles will still be (n-2)*180.
Let’s look at a question to see the application of this formula in irregular polygon scenario.
Question: The measures of the interior angles in a polygon are consecutive odd integers. The largest angle measures 153 degrees. How many sides does this polygon have?
A) 8
B) 9
C) 10
D) 11
E) 12
Solution:
The interior angles are: 153, 151, 149, 147 … and so on.
Now there are two ways to approach this question – one which is straight forward but uses algebra so is time consuming, another which makes you think but doesn’t take much time. You can guess which one we are going to focus on! But before we do that let’s take a quick look at the algebraic solution too.
Method 1: Algebra
Sum of interior angles of this polygon = \(153 + 151 + 149 + … (153 – 2(n-1)) = (n – 2)*180\).
If there are n sides, there are n interior angles. The second largest angle will be 153 – 2*1. The third largest will be 153 – 2*2. The smallest will be 153 – 2*(n-1). This is an arithmetic progression.
Sum of all terms = \(\frac{(First term + Last term)}{2} * n = \frac{(153 + 153 – 2(n-1))}{2} * n\)
Equating, we get \(\frac{(153 + 153 – 2(n-1))}{2} * n = (n – 2)*180\)
Solving this you get, n = 10
But let’s figure out a solution without going through this painful calculation.
Method 2: Capitalize on what you know
Angles of the polygon: 153, 151, 149, 147, 145, 143, 141, … , (153 – 2(n-1))
The average of these angles must be equal to the measure of each interior angle of a regular polygon with n sides since the sum of all angles is the same in both the cases.
Measure of each interior angle of n sided regular polygon = Sum of all angles / n = \(\frac{(n-2)*180}{n}\)
Using the options:
Measure of each interior angle of 8 sided regular polygon = 180*6/8 = 135 degrees
Measure of each interior angle of 9 sided regular polygon = 180*7/9 = 140 degrees
Measure of each interior angle of 10 sided regular polygon = 180*8/10 = 144 degrees
Measure of each interior angle of 11 sided regular polygon = 180*9/11 = 147 degrees apprx
and so on…
Notice that the average of the given angles can be 144 if there are 10 angles.
The average cannot be higher than 144 i.e. 147 since that will give us only 7 sides (153, 151, 149, 147, 145, 143, 141 – the average is 147 is this case). But the regular polygon with interior angle measure of 147 has 11 sides. Similarly, the average cannot be less than 144 i.e. 140 either because that will give us many more sides than the required 9.
Hence, the polygon must have 10 sides.
Answer (C).
03 Jun 2015, 05:20
Kudos
Bookmarks
QUESTIONS ON CIRCLES INSCRIBED IN POLYGONS
By Karishma, Veritas Prep.
Above we looked at questions on polygons inscribed in a circle. This week, let’s look at questions on circles inscribed in regular polygons. As noted earlier, it’s important to keep in mind that regular polygons are symmetrical figures. You need very little information to solve for anything in a symmetrical figure.
Question 1: A circle is inscribed in a regular hexagon. A regular hexagon is inscribed in this circle. Another circle is inscribed in the inner regular hexagon and so on. What is the area of the tenth such circle?
Statement I: The length of the side of the outermost regular hexagon is 6 cm.
Statement II: The length of a diagonal of the outermost regular hexagon is 12 cm.
Solution: Thankfully, in DS questions, we don’t need to calculate the answer. We just need to establish the sufficiency of the given data. Note that we have found that there is a defined relation between the sides of a regular hexagon and the radius of an inscribed circle and there is also a defined relation between the radius of a circle and the side of an inscribed regular hexagon.
When the circle is inscribed in a regular hexagon,
Radius of the inscribed circle = \(\frac{\sqrt{3}}{2}\)* Side of the hexagon
When a regular hexagon is inscribed in a circle,
Side of the inscribed regular hexagon = Radius of the circle
So all we need is the side of any one regular hexagon or the radius of any one circle and we will know the length of the sides of all hexagons and the radii of all circles.
Statement I: The length of the side of the outermost regular hexagon is 6 cm.
If length of the side of the outermost regular hexagon is 6 cm, the radius of the inscribed circle is \((\sqrt{3}/2)*6 = 3\sqrt{3}\) cm
In that case, the side of the regular hexagon inscribed in this circle is also \(3\sqrt{3}\) cm. Now we can get the radius of the circle inscribed in this second hexagon and go on the same lines till we reach the tenth circle. This statement alone is sufficient.
Statement II: The length of a diagonal of the outermost regular hexagon is 12 cm.
Note that a hexagon has diagonals of two different lengths. The diagonals that connect vertices with one vertex between them are smaller than the diagonals that connect vertices with two vertices between them. Length of AC will be shorter than length of AD. Given the length of a diagonal, we do not know which diagonal it is. Is AC = 12 or is AD = 12? The length of the side will be different in the two cases. So this statement alone is not sufficient.
Answer (A). This question is discussed HERE.
Keep in mind that you don’t actually need to solve for an answer is DS; in fact, in some questions you will not be able to solve for the answer under the given time constraints. All you need to do is ensure that given unlimited time, you will get a unique answer.
Question 2: Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?
(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π
Solution: First let’s recall that squares and circles are symmetrical figures. The given figure is symmetrical.
We don’t know any formula that will help us get the area of the curved shaded grey shape in the center. In such cases, very often what you need is to find the area of one region and subtract the area of another out of it. Here, if we subtract the area of the four circles out of the area of the square, the leftover area includes the shaded region but it includes other regions (around the corners etc) too. This is where symmetry helps us.
Notice that we can split the figure into four equal regions to get four smaller squares. Now focus on the diagram give below which shows you one such smaller square. The area around the four corners of the smaller squares is equal i.e. the area of the red region = area of the blue region = area of the yellow region = area of the green region.
Our shaded grey region has four such equal areas so
Area of the shaded grey region = Area of the smaller square – Area of one circle
Area of the shaded grey region = \((10)^2 – \pi(5)^2 = 100 – 25\pi\)
Answer (C). This question is discussed HERE.
Attachment:
GeometryPost12Ques1.jpg [ 9.24 KiB | Viewed 98176 times ]
Attachment:
GeometryPost12Ques2.jpg [ 7.89 KiB | Viewed 86393 times ]
Attachment:
GeometryPost12Ques2Fig2.jpg [ 8.58 KiB | Viewed 86307 times ]
Attachment:
GeometryPost12Ques2Fig3.jpg [ 3.78 KiB | Viewed 87339 times ]
03 Jun 2015, 04:26
Kudos
Bookmarks
INSCRIBING POLYGONS AND CIRCLES
By Karishma, Veritas Prep.
Above we looked at regular and irregular polygons. Now, let’s try to understand how questions involving one figure inscribed in another are done. The most common example of a figure inscribed in another is a polygon inscribed in a circle or a circle inscribed in a polygon. Let’s see the various ways in which this can be done.
To inscribe a polygon in a circle, the polygon is placed inside the circle so that all the vertices of the polygon lie on the circumference of the circle.
There are a few points about inscribing a polygon in a circle that you need to keep in mind:
– Every triangle has a circumcircle so all triangles can be inscribed in a circle.
– All regular polygons can also be inscribed in a circle.
– Also, all convex quadrilaterals whose opposite angles sum up to 180 degrees can be inscribed in a circle.
There are also a few points about inscribing a circle in a polygon that you need to keep in mind:
– All triangles have an inscribed circle (called incircle). When a circle is inscribed in a triangle, all sides of the triangle must be tangent to the circle.
– All regular polygons have an inscribed circle.
– Most other polygons do not have an inscribed circle.
A simple official question will help us see the relevance of these points:
Question: Which of the figures below can be inscribed in a circle?
(A) I only
(B) III only
(C) I & III only
(D) II & III only
(E) I, II & III
Solution: I think it will suffice to say that the answer is (C).
This question is discussed HERE.
Attachment:
GeometryPost8Fig1.jpg [ 11.59 KiB | Viewed 88795 times ]
Attachment:
GeometryPost8Fig2.jpg [ 11.72 KiB | Viewed 88717 times ]
Attachment:
GeometryPost8Fig3.jpg [ 9.82 KiB | Viewed 88755 times ]
Attachment:
GeometryPost8Fig4.jpg [ 11.59 KiB | Viewed 88775 times ]
Attachment:
GeometryPost8Fig5.jpg [ 14.3 KiB | Viewed 88654 times ]
Attachment:
GeometryPost8Fig6.jpg [ 11.07 KiB | Viewed 92326 times ]
