Geometry

Mistake 20

A certain right triangle has sides of length x, y, and z, where x < y < z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y? x and y are the legs of the triangle and z is the hypothenuse xy/2 = 1 xy = 2 x = 2/y Because x must be smaller than y, we can create the following inequality 2/y < y 2 < y^2 sqr(2) < y

Mistake 22

A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip? See picture on how the diagram would look like Area of brass square/ area of wooden strip = 25 /39 lets say length of the wooden plaque= y and length of the square brass = x then x^2 / (y^2 - x^2) = 25/39 x^2 = 25 y^2 = 64 x / y = 5/8 Width of wooden strip should be y-x x = 5y/8 y - 5y/8 = 3y/8 Now 3y/8 could be any value depending on the value of y, so the width can take any positive value

Mistake 30

A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius R, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and square regions in terms of R. The circle will us 2πR from the wire Hence, the square will be made with 40 - 2πR Since 40 - 2πR represents the perimeter, in order to get one side we will need to (40 - 2πR)/4 10 - (πR/2) The area of a square is side^2 (10 - (πR/2))^2 The are of a circle is πR^2 Hence the sum of the areas in terms of R is πR^2 + (10 - (πR/2))^2

Mistake 29

A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees? (1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm2, 4 cm2, and 6 cm2 (2) c < a + b < c + 2 In a right triangle the sum of the square of the legs is equivalent to the square of the hypothenuse a^2 + b^2 = c^2 If the the squared legs of the triangle sum to a different number, then we can conclude that a measure is NOT 90 degrees, and hence we will be able to answer the question Statement 1 Since we could calculate the sides of a triangle, we can proof the point mentioned above and we will have enough information Sufficient Statement 2 Not sufficent

Mistake 13

An open box in the shape of a cube measuring 50 centimeters on each side is constructed from plywood. If the plywood weighs 1.5 grams per square centimeter, which of the following is closest to the total weight, in kilograms, of the plywood used for the box? (1 kilogram = 1,000 grams) Surface area of a cube without a box = 5 x side^2 Surface area = 5 x 50^2= 12500 Since the box is made of plywood that weights 1.5m grams per square centimiter, the box will weight 12500 x 1.5 = 18750 grams Aprox 19kg

Mistake 6

Each of 27 white 1-centimeter cubes will have exactly one face painted red. If these 27 cubes are joined together to form one large cube, as shown above, what is the greatest possible fraction of the surface area that could be red? Surface area that could be red / Total surface area Total surface area: 6s^2 = 6x3^2 = 6 x 9 = 54 The edge/side of the cube is worth 3 because it is composed of 3 smaller cubes worth 1 each Surface area that could be red: Since there are 27 cubes, and there is one that is not visible (the one at the center), the upmost number of faces painted in red is 26 Hence 26/54 = 13/27

Mistake 3

If a rectangle of area 24 can be partitioned into exactly 3 nonoverlapping squares of equal area, what is the length of the longest side of the rectangle? Since the rectangle can be partitioned into 3 squares, the length of the rectangle will be 3 times the width If x = width of rectangle, 3x = length of the rectanglke X x 3X = 24 3X^2=24 X = sqr(8) X = 2sqr(2) 3X = 6sqr(8) Since we are

Mistake 12

If the diameter of a circular skating rink is 60 meters, the area of the rink is approximately how many square meters? Since the diameter is 60, the radius is 30, and the area is 30^2 x π equivalent to roughly 2800

Mistake 1

In a certain playground, a square sand box rests in a circular plot of grass so that the corners of the sandbox just touch the edge of the plot of grass at points W, X, Y and Z, as shown. What is the distance from point W to point Y? (1) The area of the circular plot is 49π (2) The ratio of the area of the sand box to the area of the circular plot is 2 / π To find the distance from point W to point Y, the statements need to provide either the length of a side of the square or the length of the diameter of the circle. Evaluate the statements one at a time. I didn't know that the length of the diameter of the circle was equivalent to the length of one of the sides of the square 1) Evaluate Statement (1). It states that the area of the circular plot is 49π. The formula for the area of a circle is A = πr², so 49π = πr² and the radius, r = 7. Thus, the diameter, WY, is 2(7) = 14. Since Statement (1) provides one specific answer to the question, which is 14, the statement is sufficient. Write down AD. 2) Now, evaluate Statement (2). It states that the ratio of the area of the sandbox to the area of the circular plot is . Since the formula for the area of a square is A = s², the information from the statement can be expressed in the equation . Cross multiply to find that s²π = 2πr² which simplifies to s² = 2r² . Since there are two variables and only one equation, Statement (2) is insufficient.

Mistake 8

In a triangle, the sum of the length two sides must be greater than the length of the third side. Also, the difference between two sides must also be less than the length of the third side Applied to this problem: x must be less than 4 + 3 x must be more than 4-3 Hence 1 < x < 7

Mistake 17

In pentagon PQRST, PQ= 3, QR = 2, RS = 4, and ST = 5. Which of the lengths 5, 10, and 15 could be the value of PT ? Two ways to solve it 1) The length of any side of a triangle must smaller than the sum of the other two sides. The same for pentagon: the length of any side of a pentagon must be smaller than the sum of the other four sides.PQ + QR + RS + ST = 3 + 2 + 4 + 5 = 14, so the length of the fifths side can not be more than 14 2) You can break down pentagon into 3 triangles and create inequalities to get to the maximum length of side PT

Mistake 31

In the figure above, if MNOP is a trapezoid and NOPR is a parallelogram, what is the area of triangular region MNR? (1) The area of NOPR is 30 (2) The area of the shaded region is 5 Things that I need to notice: 1. Triangle MNR is an isosceles triangle and MN=NR 2. Since NOPR is a parallelogram, NO=RP and NR=OP 3. Combining 1 and 2, MN=NR=OP and NO=RP 4. Since NR=OP, angle in P will be equal than angle x, and since triangle MNR is isosceles, we can break down the triangle into 2 right triangles From the 4th point, we can deduce that the area of triangle MNR is twice the are of triangle QOP. Hence, if we know the are of triangle QOP, we have the area of MNR Statement 1 Insufficient Statment 2 Area of QOP=5 Area of MNR=10 Sufficient

Mistake 5

In the figure above, if triangles ABC, ACD, and ADE are isosceles right triangles and the area of ΔABC is 6, then the area of ΔADE is Because triangles are isosceles right triangles, they are 45-45-90 triangles with a ratio of x : x : xsqr(2) AB = x AC = xsqr(2) DC = xsqr(2) DA = x (sqr(2)) = x sqr(2) x (sqr(2)) = 2x DE= 2x Hence the area of ADE is (2x x 2x/2) = 4x^2/2 = 2x^2 Since we know that AB x BC or X x X = 12 x^2 = 12 Hence 2(12) = 24

Mistake 21

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x ? Go to link to see answer: https://gmatclub.com/forum/in-the-figure-above-point-o-is-the-center-of-the-circle-and-oc-ac-107874.html

Mistake 15

In the figure shown, a square grid is superimposed on the map of a park, represented by the shaded region, in the middle of which is a pond, represented by the black region. If the area of the pond is 5,000 square yards, which of the following is closest to the area of the park, in square yards, including the area of the pond? Since the pond amounts to 5000 square yards, one square amounts to 2500 square yards. Since there are a total of 30 squares, the total area of the map is 30 x 2500 = 75000 Hence, the area of the park must be less than 75,000 Roughly we can count 10 squares that don't belong to the park 10 x 2500 = 25,000 Hence the area of the park is 75,000 - 25,000 = 50,000

Mistake 35

In the figure shown, line segments QS and RT are diameters of the circle. If the distance between Q and R is 8/√2, what is the area of the circle? The distance between two points is the shortest distance between then. It is not the arc!! We can draw a right triangle with hypothenuse of 8/√2 and two legs that are equivalent, since they are the radii of the circle The triangle is a 45-45-90 triangle with ratio of x - x - xsqr(2) Since we know the value of sqr(2), we can create the followinf equation xsqr(2) = 8/√2 x = 4 The radii of the circle is 4 The are of the circle is πr^2= 16π

Mistake 32

In the figure shown, what is the area of triangular region PRT ? (1) The area of rectangular region PQST is 24. (2) The length of line segment RT is 5. We can calculate the area of the triangle by multiplying lenght and the width of the rectangle. Statement 1 Gives us lxw. The area of the triangle is 12 Statement 2 Not sufficient

Mistake 16

In the figure shown,PQRSTU is a regular polygon with sides of length x. What is the perimeter of triangle PRT in terms of x? Since the figure is a regular hexagon, the sum of the interior angles is (n-2)x180 where n= 6 Hence each angle of the hexagon is worth 120 degrees Since one angle is worth 120 degrees, we can draw a line that bisects the obtuse angle to create a 30-60-90 angle. 30 - 60 - 90 angles are in the ratio of x : xsrq(3) : 2x Since the hypothenuse (2x) is worth x, the smaller leg is worth x/2 and the larger leg is worth xsqr(3)/2 Since the larger leg is just half of one side of the inscribed figure, the side must be 2x that, hence Side of the inscribed figure xsqr(3)/2 x 2= xsqr(3) Since the figure has 3 sides, the perimeter is 3x sqr(3)

Mistake 28

In triangle XYZ, what is the length of YZ? (1) The length of XY is 3. (2) The length of XZ is 5. The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. (1) The length of XY is 3. Not sufficient. (2) The length of XZ is 5. Not sufficient. (1)+(2) (5-3) < YZ < (5+3) --> 2 < YZ < 8. Not sufficient.

Mistake 2

Is the angle M = 90 degrees? (1) The length of arc LMN is half the circumference of the circle. (2) LN is the diameter of the circle. Evaluate Statement (1), which states that the length of arc LMN is equal to half the circumference of the circle. Therefore, LN is the diameter of the circle. If a triangle is inscribed in a circle with one of its sides the diameter of the circle, then the triangle is a right triangle. Thus, the answer to the question is "Yes" because . Since Statement (1) produces a consistent "Yes" answer to the question, the statement is sufficient. So, write down AD. Now, evaluate Statement (2), which states that LN is the diameter of the circle. If a triangle is inscribed in a circle with one of its sides the diameter of the circle, then the triangle is a right triangle. Thus, the answer to the question is "Yes" because . Since Statement (2) produces a consistent "Yes" answer to the question, the statement is sufficient. Eliminate choice A. The correct answer is choice D.

Mistake 26

It costs $2,250 to fill right circular cylindrical Tank R with a certain industrial chemical. If the cost to fill any tank with this chemical is directly proportional to the volume of the chemical needed to fill the tank, how much does it cost to fill right circular cylindrical Tank S with the chemical? (1) The diameter of the interior of Tank R is twice the diameter of the interior of Tank S. (2) The interiors of Tanks R and S have the same height. The volume of a cylinder = πr^2h, and we are given that the price is directly proportional to the volume. Hence, if we know the relationship of the volume between tank R and tank S, we will be able to calculate the price of tank S. Statement 1 We know the radius relationship but no the height. Not sufficient Statement 2 We know the height relationship but not the radius. Not sufficient Both statements combined We know both the radius and height relationship, we can calculate the price of tank S

On objects where there is no volume i.e a circle, a square, a rectangle, the surface area refers to the actual area

On objects where there is volume i.e a cube, a right circular cylinder, the surface area refers to the surface that wraps up that object

Mistake 23

One side of a parking stall is defined by a straight stripe that consists of n painted sections of equal length with an unpainted section 1/2 as long between each pair of consecutive painted sections. The total length of the stripe from the beginning of the first painted section to the end of the last painted section is 203 inches. If n is an integer and the length, in inches, of each unpainted section is an integer greater than 2, what is the value of n ? We can define x as the length of one painted section We can define x/2 as the length of the space between painted sections Since there are n painted sections, the total length of the painted sections is nx The space in between is a bit more tricky. Since one space only arises when there are two painted sections, then a space is equal to n-1 Hence the equation is as follows: nx + x(n-1)/2 = 203 2nx + nx - x = 406 3nx - x = 406 x(3n-1) = 406 x(3n-1) = 2 x 7 x 29 Since n must be an integer, we know that n must be an integer multiple of 3 minus 1. Out of the possible choices only 2, and 29 satisfy the criteria of integer multiple of 3 minus 1. If 3n-1= 2 3n=3 n= 1 Not in the answer choice and the number of spaces would be 0 so it' not a possible answer if 3n-1= 29 3n= 30 n= 10 There are a total of 10 painted spaces

Mistake 4

Point X lies on side BC of rectangle ABCD, which has length 12 and width 8. What is the area of triangular region AXD ? Look at how the triangle is inscribed into the rectangle The area of the triangle will be (bxh/2) b= 12 h= 8 (12x8)/2 = 48 The mistake that I made here is that I didn't know how to illustrate point x and the new triangle

Mistake 33

Points M and P lie on square LNQR, and LM = PQ. What is the length of the line segment PQ? 1) PR= 4sqr(10)/3 2) The ratio of the area of the unshaded region to the total area of the shaded region is 2 to 1. Statement 1 Since we know the value of PR and QR, we can calculate the value of PQ using the Pythagoras Theorem Statement 2 We know that the area is 16, because the rectangle is a square with sides 4 Unshades/Shades = 2x / 1x and the sum of the areas is 2x + 1x = 3x Hence, 16 = 3x Since the area of the unshaded region is (base x height)/2 and we have the area, and the base(side), we only need to calculate the height in order to get the value of PQ

Mistake 7

The figure above is constructed by separating a circular region into 6 equal parts and rearranging the parts as shown. If the diameter of the circle is d, what is the perimeter of the figure above? What I have to notice here is that the circumference (dπ) is split in 6, and hence each rounded edge is worth dπ/6 On top of that because the solid lines are equivalent to the radius (d/2), the corners are equivalent to d/2, therefore summing both parts we obtain dπ + d As per GMAT club: Perimeter = d/2 + d/2 + (1/6 of circumference) + (1/6 of circumference) + (1/6 of circumference) + (1/6 of circumference) + (1/6 of circumference) + (1/6 of circumference)

Mistake 27

The figure above shows Line L, Circle 1 with center at C1, and Circle 2 with center at C2. Line L intersects Circle 1 at points A and B, Line L intersects Circle 2 at points D and E, and points C1 and C2 are equidistant from line L. Is the area of ΔABC1 less than the area of ΔDEC2 ? (1) The radius of Circle 1 is less than the radius of Circle 2. (2) The length of chord AB is less than the length of chord DE. The area of a triangle is 1/2 x base x height. Since both triangles are equidistant to the line, we know that the heights of the triangles is the same. Hence, what the question is really asking is: Is the base of triangle ΔABC1 < base of ΔDEC2? Or: AB<DE? Statement 1 Since the radius of C1 < C2, then a chord that crosses C1 must be less than. a chord that crosses C2. For this reason AB<DE. Sufficient Statement 2 AB<DE. Sufficient

Mistake 24

The figure above shows a piece of cheese with a corner cut off to expose plane surface ABC. What is the area of surface ABC ? (1) AD = 10 centimeters (2) The shape of the cheese was a cube before the corner was cut off. The Logical approach to this question is quite simple: we must have at least one number to find the area of the triangle, and thus we need statement (1), but it is not sufficient on its own. In statement two were told that the triangle is an equilateral triangle, since a cube is symmetrical. We only need one side of an equilateral triangle in order to find its area, and thus both statements together are required, and the correct answer is (C). The Precise approach would use congruence of triangles and the formula for the area of equilateral triangle, but, as in most DS questions, that would be longer and redundant as we're not asked what the area IS, but rather whether there's enough information to solve the question.

Mistake 25

The figure above shows a portion of a road map on which the measures of certain angles are indicated. If all lines shown are straight and intersect as shown, is road PQ parallel to road RS? (1) b = 2a (2) c = 3a For two lines to be parallell, they must be intersected by a straight line that in turns creates a 90 degree angle in both lines. In this case, since the line crossing both lines is straight, the lines will be parallel if and only if angle C is 90 degrees. Hence, we need to find a way to find angle C. Statement 1 We can't get angle C. Not sufficient Statement 2 We can't get angle C. Not sufficient Both statements combined We can get angle C, and it's measure is 90 degrees. Hence the lines are parallel

Mistake 9

The figure above, which is divided into 6 sectors of equal area, contains an arrow representing a spinner. If the spinner is rotated 3,840 degrees in a clockwise direction from the position shown, which of the following indicates the sector to which the arrow on the spinner will point? If the arrow rotated 3600 degrees, then it would have spinned 10 times. Since the arrow has spinned 3840 degrees, a 240 degree difference, then the arrow has advanced. Since each region is worth 60 degrees 360/6, then we can calculate where the arrow sit by 240/60 = 4 The arrow has moved 4 sectors clockwise As per GMAT club: The arrow spinner travels 360 degrees for every complete revolution.In 10 revolutions, the arrow spinner completes 360 x 10 = 3600 degrees.The remaining 240 degrees is left. Also, note that each of the sectors holds 60 degrees.240 = 60 x 4. Therefore, the arrow spinner travels exactly 4 sectors from its current position.That is, the arrow spinner reaches the midway of the sector containing the square.

Mistake 11

The figures above show a sealed container that is a right circular cylinder filled with liquid to 1212 its capacity. If the container is placed on its base, the depth of the liquid in the container is 10 centimeters and if the container is placed on its side, the depth of the liquid is 20 centimeters. How many cubic centimeters of liquid are in the container? The question is asking us what is half the volume of the right circular cylinder from the picture From the picture we can derive that R= 20 and H= 20 Since the formula for the volume of a right circular cylinder is π x r ^2 x h (π x 20^2 x 20)/2 = 4000π

Mistake 14

The perimeter of rectangle A is 200 meters. The length of rectangle B is 10 meters less than the length of rectangle A and the width of rectangle B is 10 meters more than the width of rectangle A. If rectangle B is a square, what is the width, in meters, of rectangle A ? Length A = x Width A = y Length B = p Width B = r 2x + 2y = 200 p = x - 10 r = 10 + y r = p x - 10 = 10 + y x - 20 = y 2x + 2y = 200 x + y = 100 x = 100 - y 100 - y - 20 = y 80 = 2y y = 40

Mistake 19

The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4π/3, what is the length of line segment RU? Radius= 4 Although we don't have any diagram for reference, we can create our own. See the image The circumference of a circle is 8π. Since arc RTU measures 4π/3, it accounts for 4π/3/8π of the circle, equivalent to 1/6 So, the arc RTU is 1/6 th of the circumference. This means that ∠RCU=360/6= 60 degrees (C center of the circle). RCU is isosceles triangle as RC=CU=r and ∠RCU=∠CRU=∠CUR=60. Hence RU=r=4

Mistake 10

When a rectangular vat that is 3 feet deep is filled to 2/3 of its capacity, it contains 60 gallons of water. If 7.5 gallons of water occupies 1 cubic foot of space, what is the area, in square feet, of the base of the vat? Volume rectangle = l x w x h Volume rectangle = h x area of the base The area of the base is length x width height = 3 Since the capacity of the rectangle is 60 gallons when it's 2/3 full, the entire capacity is 90 gallons i,e the volume of the rectangle is 90 Volume rectangle = 90/7.5= 12 12 = 3 x area of the base Area of the base = 4

Mistake 18

When the figure above is cut along the solid lines, folded along the dashed lines, and taped along the solid lines, the result is a model of a geometric solid. This geometric solid consists of 2 pyramids each with a square base that they share. What is the sum of number of edges and number of faces of this geometric solid? The best approach here is to ignore the picture From what the prompt describes, the final figure is a dual pyarmid, that goes up and down connected by a square Since a pyramid has 4 faces, the total number of faces will be 8 Since a pyramid has 4 edges, the total number of edges will be 8 Finally, we need to account for the edges of the square which are 4. Hence total faces + total edges = 16 + 4 = 20

Mistake 34

Which of the sides of ∆ PQR is the longest? (1) PQ is longer than QR . (2) The measure of ​​ is 70°. RULE - the longest side of a triangle has the largest angle opposite to it.. Statement 1 (1) PQ is longer than QR.we know QR is not the answer..But we do not know about the third side PR Insufficient Statement 2 (2) The measure of angle PQR is 70 degrees. Wwe know from this that the other two angles combined are 180-70=110..largest side has the largest angle against it..It is possible that both angles are less than 70 : 60 and 50Also it is possible that one angle is more than 70 : 75 and 35 Insufficient Both statements combined Combined we still cannot say if PQ is longest or PR Insufficient