GMAT: Geometry

Quadrilateral

area= b x h all interior angles add up to 360 degrees

Pythagorean Theorem- right triangle

a² + b² = c²

Sides of a 45, 45, 90 right triangle (isosceles right)

x, x, x√2 Has a right angle ( (90°), and also two equal angles

Sides of a 30,60,90 right triangle

x, x√3, 2x

The length of an arc of a circle

2πr∗(θ/360°) where θ is the central angle for that arc.

Right triangle ratio sets (AKA Pythagorean Triples)

3:4:5, 5:12:13, 6:8:10, 7:24:25, 8:15:17, 9:40:41

"Perimeter" of a Circle AKA

= Circumference of a circle = 2πr

Perimeter of a triangle

Distance around the edge of a triangle

What is the greatest distance between two points in a cylinder where the area of the base is 9π and the height is 5? A) √34 B) √48 C) √61 D) √76 E) √106

For these three-dimensional, greatest-distance problems, *recognize that you're almost always looking for a right triangle.* So here you just need to find the diameter, because they give you the height. Since area of a circle is πr2, and the area is 9π, then the radius is 3, so the diameter is 6. Now you have your right triangle: base of 6 and height of 5. Plugging this into Pythagorean theorem: 62+52=c2, so c2=61. Therefore, C is the correct answer.

A rectangular flat sheet of metal with length 15 and width 12, is to be converted into an open box. To do this, four squares with sides x must be cut off. What value of x would yield a box with the greatest volume? A) 5 B) 4 C) 3 D) 2 E) 1

For this problem, the first important step is to recognize the dimensions of each box with regard to x. x will become the height of the box, with the current side of 12 losing two x's (one on each end) and the current side of 15 also losing two x's (again, one on each end). So your volume formula for each potential box is: x (12 - 2x) (15 - 2x). Now plug n play from your answer choices. A: x = 5, volume = 5 * 2 * 5 = 50 B: x = 4, volume = 4 * 4 * 7 = 112 C: x = 3, volume = 3 * 6 * 9 = 162 D: x = 2, volume = 2 * 8 * 11 = 176 E: x = 1, volume = 1 * 10* 13 = 130 D has the largest volume, and is therefore correct.

Isosceles triangle

Isosceles Triangle Two equal sides Two equal angles

(38) Which of the following points falls outside of the region defined by 2y≤6x−12? A) (5, 1) B) (3, -8) C) (2, 0) D) (1, -2) E) (0, -14)

Plug & Play or make a plot line If you plug & play, first, simplify the equation by dividing 2 out. Then, plug in the #'s and you will find that D does not satisfy the equation.

There are 2 points on a graph (0,3) (6,0) that create a line. If the x-coordinate of point E on the line is 4, what is its y-coordinate? A) -1/2 B) 1 C) 3/2 D) 2 E) 7/2

Solution: B Given two points, determine the equation for the line. The slope is (0-3)/(6-0)= -1/2, and the y intercept is 3, so the equation is y = (-1/2)x+3. Now just plug in x = 4 and see what shakes out. y = (-1/2)(4)+3 = -2+3=1.

A large cube is made up of smaller equally sized cubes, where each smaller cube has sides 1/3 the length of the large cube. What approximate percent of the total volume is one of the small cubes? A) 3 B) 3.7 C) 9 D) 11.11 E) 33.33

Solution: B Similarity renders this question trivially easy. The volume of any figure is a cubic unit, so volume always scales by the cube of the linear scaling factor between similar figures. All cubes are similar to one another, and the smaller cubes here scale by a linear factor of 1/3, so the smaller cubes must each have (1/3)3=1/27 of the volume of the larger cube. 1/27 is below 4% (1/25) but above 3.33% (1/33), so the only possible answer is B.

Which of the following points is exactly halfway between (3, 9) and (5, -3)? A) 4, 6 B) 4, 3 C) 6, 4 D) 2, 12 E) 2, 6

Solution: B To get the midpoint, just average each coordinate in turn. The x coordinates are 3 and 5, and their average is 4. The y coordinates are 9 and -3, and their average is 3. The midpoint is (4,3). The correct answer is B.

If the area of triangular region RST is 25, what is the perimeter of RST? 1) The length of one side of RST is 5√2 2) RST is a right isosceles triangle.

Solution: B Leverage your assets. Statement 1 is not sufficient, as it only provides the length of one side. Statement 2, however, is sufficient. It requires you to blend two important concepts about *right* triangles: 1) that given the right angle you can use the two short sides to calculate the area using ½ bh. 2) that in a right isosceles triangle, a = b. therefore, in this triangle, ½ a²=25 (since a is both base and height). And once you solve for a, you'll be able to multiply it by √2 to find the hypotenuse.

Find the distance between points (21, -5) and (30, 7) in the coordinate geometry plane? A) 10 B) 12 C) 15 D) 18 E) 35

Solution: C *The distance formula is really just the Pythagorean Theorem in disguise* The distance between the two points in the x direction is 30 - 21 = 9. The distance between the two points in the y direction is 7 - (-5) = 12. Students should recognize the triple 9, 12, 15, which is just 3, 4, 5 scaled up by a factor of 3 (or you can square, add, and work it out). So the distance between (21, -5) and (30, 7) is 15.

Rex has a 24 ft2 sheet of wood and cuts it into 6 identical square pieces (with no pieces of wood left over). He uses these pieces to make a box. How much dirt can this wood box hold (in ft3)? A) 1 B) 4 C) 8 D) 16 E) 24

Solution: C In order for the six identical square pieces to produce a total area of 24 ft², each square piece individually must have an area of 24/6=4 ft². The area of a square is side2, so each square piece must have side length 2 ft. Finally, putting the squares together makes a cubical box with edge length 2 ft, and the volume of a cube is edge3, so this box will have volume (2ft)3=8ft³.

A dog is tied to the corner of a fence with a 9 foot chain. If the angle of that corner of the fence is 120 degrees, how many square feet does the dog have to walk around (the dog is tied inside, not outside, the fence)? A) 19π B) 27 C) 27π D) 81 E) 81π

Solution: C The dog can roam a distance of radius 9. A circle with radius 9 has area πr²=π∗9²=81π, so this a find the length, we multiply 81π(120/360) = 81π(1/3) = 27π.

What fractional part of the total surface area of cube C is red? 1) Each of 3 faces of C is exactly 1/2 red. 2) Each of 3 faces of C is entirely white.

Solution: C To calculate the fraction of the whole cube that's red, we need to know what fraction of each of the six faces is red. Statement (1) only discusses three of the faces, leaving the other three faces unknown, so it's insufficient. Statement (2) only discusses three of the faces, leaving the other three faces unknown, so it's insufficient. Together, the statements provide information about all six faces: three are half red, and the other three contain no red at all (they're entirely white). Thus, the two statements together are sufficient to establish the fractional part of the total surface area of cube C that is red. The correct answer is C.

A three-dimensional "skeleton" rectangular shape is made up of metal rods. The rods' total length is 480. The base is a square with sides of length x. Which of the following expresses that height? A) 960 - 16x B) 480 - 8x C) 240 - 4x D) 120 - 2x E) 60 - x

Solution: D Labeling the height as h (label your unknowns!) and counting up the rods, there are 8 rods with length x (four on top and four on bottom) and 4 rods with length h. So 8x+4h=480, and we can solve for the height. 4h = 480 - 8x; h = 120 - 2x. The answer is D.

A circle has an area of y. What is the length of its diameter in terms of y? A) √(y/π) B) √(π/y) C) √(2y/π) D) 2√(y/π) E) 2√(π/y)

Solution: D The area y=πr². Let's solve for r: r²=(y/π), so r=√(y/π). The diameter is d=2r, so d= 2√(y/π).

A soft-drink producer has done marketing research and found that if it decreases the width of its soda cans (right circular cylinders) by 50%, but keep the height the same, they can still sell the cans for 75% of the original price. By what percent does the price per volume of soda increase to the consumer? A) 25% B) 50% C) 100% D) 200% E) 300%

Solution: D The original volume of the soda cans is V=πr²∗h, and the original price is P. The new volume is π(r/2)²∗h=(1/4)πr²∗h=(1/4)V, and the new price is (34)P. The question asks about price per volume, which was, originally, PV, and is, finally, (34)P/((14)V)=4∗(34)PV=3PV. The price per volume has tripled! The temptation here is to jump hard on answer choice E, but wait just a minute and re-read the question. The new price is 300% of the original price, but the question asks for the *percent change* Using the percent change formula % Change = ((New−Old)Old) * 100%, we have %Change = ((3PV−PV)/(PV)) * 100% =((2PV)/(PV)) * 100% = 2 * 100% = 200%.

A deli sells soup, priced by weight, in two sizes of right circular cylindrical plastic containers. The height and the radius of size A are each three times that of size B. If a customer fills a size B container of soup to capacity, and pays $2, how much would a size A container filled to half of its capacity cost? A) $9 B) $12 C) $18 D) $21 E) $27

Solution: E Because the heights and radii of the two containers scale up by the same factor, the two containers are similar, and we can use similarity to solve this question without even calculating the volume of either cylinder. The volume of any figure is a cubic unit, so volume always scales by the cube of the linear scaling factor between similar figures. All lengths of container A are three times those of container B, so the volume of container A must be 3³ = 27 times the volume of container B. When half full, container A will hold 27/2 times as much soup as will container B. Since container B, full to capacity, costs $2, container A, half full, must cost (27/2) * $2 = $27.

In three-dimensional space, if each of the two lines L1 and L2 is perpendicular to line L3, which of the following must be true? (1) L1 is parallel to L2. (2) L1 is perpendicular to L2. (3) L1 and L2 lie on the same plane. A) I only B) I and II C) II and III D) III only E) none of the above

Solution: E None of the above must be true. Imagine L3 as a line from left to right across the page. Now, imagine that L1 is vertical, crossing L3 at a right angle: Finally, imagine that L2 goes into the page, in three-dimensional space, and intersects L3 perpendicularly at a different point. In this case, none of the listed statements are true, so the answer must be E. Note that statement III in this case was essentially a hint. Even if you had thought points I and/or II were valid, consideration of statement III should have inspired the question "what if the lines don't share a plane?" In that case, surely, L1 and L2 would not be either parallel or perpendicular to one another.

A rectangle is defined to be "silver" if and only if the ratio of its length to its width is 2 to 1. If rectangle S is silver, is rectangle R silver? 1) R has the same area as S. 2) The ratio of one side of R to one side of S is 2 to 1.

Solution: E Statement (1) alone is not sufficient to answer the question because R could have the same dimensions as S (e.g., 4 : 2) and be silver, or R could have different dimensions (e.g., 8 : 1) and not be silver. Thus the answer is B, C, or E. Statement (2) alone does not tell anything about the relationship between the other sides of R and S, and so it is not sufficient; the answer must be C or E. The logic applied to (1) can also be applied to the information given in 2); thus (1) and (2) together are not sufficient, and the answer is E.

The shaded portion of a rectangular yard is triangle ABC. If the area of the patio is 30 square feet and b = a + 7, then c equals _____? A) √2 B) √22 C) 5 D) 12 E) 13

Solution: E The area of a triangle is (1/2)bh, so the patio in this case has area (1/2)ab. We're told that this area is 30, so (1/2)ab=30 and ab=60. It's also given that b=a+7, so a(a+7)=60 and a2+7a-60=0. Factoring, (a+12)(a-5)=0, and a=5 or -12. Since it's a length, a=5. b=a+7=5+7=12. Finally, by the Pythagorean Theorem, c2=52+122=25+144=169, and c=13. (5,12,13 is a Pythagorean Triple worth knowing for the GMAT!) The answer is E

What is the perimeter of triangle PQR? 1) The measures of angle PQR, QRP and RPQ are x°, 2x° and 3x°. respectively. 2) The altitude from triangle PQR from Q to PR is 4.

Statement (1) enables the calculation of all angle measures. It's not actually necessary to calculate the angles, but the way to do it is to recall that the angles of a triangle add up to 180°, so x°+2x°+3x°=180°, 6x°=180°, and x°=30°. It's a 30°-60°-90° right triangle. Regardless, angle-angle-angle doesn't determine any lengths in the triangle - it only establishes similarity. Statement (1), standing alone, is insufficient. Statement (2) is pretty useless by itself. Knowing only the height establishes virtually nothing about the shape or perimeter of the triangle. Putting both statements together, however, we know all angles of the triangle, which establishes similarity, and one length. Given similarity, any one length fully determines everything about the figure, and "everything" includes the perimeter, so the statements taken together are sufficient. The answer is C. Once again, it's not necessary to actually calculate here, but it is certainly possible to do so. As a 30°-60°-90° right triangle, PQR has side lengths in the ratio of x:x√(3):2x. The altitude of 4 just so happens to be the x√ (3)side of the triangle, so x=4/√(3) and 2x=8/√(3) The perimeter will be 4/√(3)+8/√(3)+4. C is correct.

What is the length of minor arc line AEB with angle 30° in the circle which has a diameter of line CFD equaling 12? A) 2π B) 4π C) 6π D) 8π E) It cannot be determined with the information given.

The length of an arc of a circle is 2πr∗(θ/360°), where θ is the central angle for that arc. Only the central angle will work, though, and the diagram in this case shows an inscribed angle instead. But recall that the measure of any inscribed angle is exactly half that of the corresponding central angle. Angle AEB measures 30°, so angle AFB, otherwise known as θ, measures 60°. Furthermore, the radius of the circle is 6, so the formula gives the arc length in question as 2π∗6∗(60°/360°)=2π∗6∗(1/6)=2π. The correct answer is A.

A pizza with diameter of 12 inches is split into eight equally sized pieces. Four non-adjacent pieces are removed. What is the perimeter AOBCODEOFGOHA of the pizza now, including the inside edges of the slices? A) 48π+48 B) 24π+48 C) 24π+24 D) 6π+48 E) 6π+24

The perimeter you're after has two components - part circumference (the crust), and part radius (the distance alongside each slide, for example length AO). You know that the radius is 6 because r = ½ (d), and d = 12. And you know that the circumference is 12π. With 8 slices of pizza, then, each slice contains 1.5π of crust length. So now you need to determine how many lengths are left. 4 slices remain, so that's 6π of circumference length. And each slice has two sides of length 6, so you'll have 4 x 2 x 6 = 48 total inches of radius. Solution: D

Slope of a line

To find slope (mx) =(y₁-y₂)/(x₁-x₂) rise/run EQUATION: y= mx+ b mx=slope b=point where line intercepts y axis

If a large pizza has a radius that is 30 percent larger than that of a medium pizza, what is the percent increase in area between a medium and a large pizza? A) 30 B) 36 C) 60 D) 69 E) 90

Try to recognize the relationship between radius and area. Area = πr², so if you multiply the radius by 1.3 (to increase the area by 30%), then once you square it you now have 1.69r². That means that the area has increased by 69%, making D the correct answer.

In triangle PQR above is PQ > RP? 1) x = y 2) y = z

Using the properties of isosceles triangles, if two angles of a triangle are equal then the two opposite sides of that triangle are equal (and vice versa). Statement (1) states that x=y, and therefore QR=PQ - but this says nothing about the length of RP, so it's insufficient. Statement (2) states that y=z, and therefore PQ=RP. So the statement PQ > RP is false, and on data sufficiency "no means yes"; regardless of the fact that the answer was no, the point is that we had sufficient information answer the question. So B is the correct answer.

Volume of a rectangle

V= l x w x h

Volume of a Cylinder

V=volume of cylinder= V=(πr²)h

If the width, depth and length of a rectangle box were each decreased by 50%, by how many percent would the volume of the box decrease? A) 12.5% B) 25% C) 50% D) 75% E)87.5%

Volume always scales by the cube of the linear scaling factor between similar figures. In this case, the box is scaled down by a linear factor of (1/2), so its volume scales down by a factor of (1/2)3 = 1/8. The percent change formula gives ((New-Old)/Old) * 100% = (((1/8)-1)/1) * 100% = (-7/8) * 100% = -87.5%. The correct answer is E.

Combined Rate Formula

X × Y / X + Y

Surface Area of a Prism

add up area of all sides, for a square (b*h)+(b*h)+(b*h)+(b*h)

Area of a triangle

½ base x height