Interested in Shapes: Geometry Let’s move: Lines and Angles The building blocks for geometric forms are lines and angles, so we start by defining these
fundamental elements. Although the definitions aren’t directly tested, understanding the meanings of these terms is an important part of solving problems on the GRE. Here are the common terms that pop up on the test: Line: A straight path of points that extends forever in two directions. A line does not have any width or thickness. Arrows are sometimes used to show that the line goes on forever.
Line segment: The set of points on a line between any two points on the line. Basically it’s just a piece of a line from one point to another that contains those points and all the points between.
Ray: A ray is like half of a line; it starts at an endpoint and extends forever in one direction. You can think of a ray as a ray of light extending from the sun (the endpoint) and shining as far as it can go.
Midpoint: The point halfway (equal distance) between two endpoints on a line segment. Bisect: To cut something exactly in half, such as when a line segment cuts another line segment or an angle or a polygon into two equal parts. Abisector is a line that divides a line segment, an angle, or a polygon into two equal parts.
Intersect: Just like it sounds — intersect simply means to cross; that is, when one line or line segment crosses another line or line segment.
Collinear: A set of points that lie on the same line. Vertical: Lines that run straight up and down. Horizontal: Lines that run straight across from left to right or right to left (you can decide later!!)
Parallel: Lines that run in the same direction, always remaining the same distance apart. Parallel lines never cross one another. Perpendicular: When two lines intersect to form a square corner. The intersection of two perpendicular lines forms a right, or 90°, angle.
Angle: The intersection of two rays (or line segments) sharing a common endpoint. The common endpoint is called the vertex. The size of an angle depends on how much one side rotates away from the other side. An angle is usually measured in degrees.
Acute angle: Any angle measuring less than 90°. Like an acute, or sharp, pain, the acute angle has a sharp point.
Right, or perpendicular, angle:An angle measuring exactly 90°. It makes up a square corner.
Obtuse angle: An angle that measures more than 90° but less than 180°. The opposite
of an acute angle, an obtuse angle is dull rather than sharp.
Straight angle: An angle that measures exactly 180°. A straight angle appears to be a straight line or line segment.
Complementary angles: Angles that add together to total 90°. Together, they form a right angle.
Supplementary angles: Angles that add together to total 180°. They form a straight angle.
Similar: Objects that have the same shape but different sizes. Congruent: Objects that are equal in size and shape. Two line segments with the same length, two angles with the same measure, and two triangles with corresponding sides of equal lengths and angles that have equal degree measures are congruent.
Lines and angles form figures, and one of the most popular GRE figures is the triangle. A tri- angle has three sides, and the point where two of the sides intersect is called a vertex. We name triangles by their vertices, so a triangle with vertices A, B, and C is called ABC. The majority of geometry questions on the GRE involve triangles, so pay particular attention to the properties and rules of triangles. Just as lines and angles have rules that apply to lots of situations, triangles have rules that apply to all triangles. But some triangles are so special that some rules exist just for them:
An isosceles triangle has two equal sides, and the measures of the angles opposite those two sides are also equal to each other.
An equilateral triangle has three sides of equal lengths and three angles of equal measure.
A right triangle has one angle that measures 90°. The side opposite the right angle is called the hypotenuse.
The measures of the three angles of any triangle always add up to 180°.
Triangles have great proportions. So the smallest angle faces the shortest side of the triangle and the largest angle faces the longest side. If two or more angles have the same measurement, their opposite sides are also equal.
Area of a triangle: A = 1/2bh. A stands for (what else) area, b is the length of the base or bottom of the triangle, and h stands for the height (or altitude), which is the distance that a perpendicular line runs from the base to the angle opposite the base.
Notice that, as shown in Figure above, the height or altitude is always perpendicular to the base and that the height can be placed either inside or outside the triangle.
Getting connected right: Right Angle Triangles
a
c
b The Pythagoras Theorem a2 + b2 = c2
Keep in mind that the Pythagorean theorem works only with right triangles. You can’t use it to find the lengths of sides of triangles that don’t have a right angle in them. The Triple Treat again:
There are a few numbers that follow the Pythagoras Theorem and have also caught the fancy of GRE. It will be handy to have these ratios based on the Pythagoras theorem in mind. That way, you don’t have to work out the whole theorem ever y time you deal with a right triangle. 32 + 42 = 52 52 + 122 = 132 62 + 82 = 102
Knowing what’s neat about the 30°:60°:90° triangle Some other handy right triangles exist. One is the 30°:60°:90° triangle. When you bisect any angle in an equilateral triangle, you get two right triangles with 30°, 60°, and 90° angles. In a 30°:60°:90° triangle, the hypotenuse is two times the length of the shorter leg,
as shown in figure.The ratio of the three sides isx : of the shortest side.
x
3 : 2x , where x = the length
Feeling the equilibrium of a 45°:45°:90° triangle If you bisect a square with a diagonal line, you get two triangles that both have two 45° angles. Because the triangle has two equal angles (and therefore two equal sides), the resulting triangle is an isosceles right triangle, or 45°:45°:90° triangle. Its hypotenuse is equal to 2
times the length of a leg. It’s important to recognize this also means that the length of a leg is equal to the length of the hypotenuse divided by right triangle is, therefore x : x : x 2
2. The ratio of sides in an isosceles
A striking resemblance: Similar triangles Triangles are similar when they have exactly the same angle measures.Similar triangles have the same shape, even though their sides are different lengths. The corresponding sides of similar triangles are in proportion to each other. The heights or altitudes of the two triangles are also in proportion. Figure provides an illustration of the relationship between two similar triangles.
Let’s try some questions: Q1. If the lengths of two sides of a triangle are 5 and 9, respectively, which of the following could be the length of the third side of the triangle? Indicate all such lengths. A) 3 B) 5 C) 8 D) 15 E) 18 Q2. What is the value of a + b?
A) B) C) D) E)
30 50 55 65 90
Q3. If l1 is parallel to l2, what is the value of x + 2y?
A) B) C) D) E)
90 120 180 270 360
Q4.
Three lines intersect in a point as shown in the figure above. Which of the following pairs of angle measures is NOT sufficient for determining all six angle measures? A) B)
t
C)
s
D)
r
E)
r
t
and z and y and and and
x
t s
Q5.
In the figure above, the dotted lines bisect the angles with measurex and y value of z, if x = 70 and y = 40?
, what is the
Q6.
In the figure above, if l || m, what does A) B) C) D) E)
z
equal in terms of
x
and y ?
x+y x–y 180 – x 180 – x + y 180 – x – y
Q7. Which of the following is a possible length for side AB of triangle ABC if AC = 6 and BC = 9? I. II.
3 9√3
III. A) B) C) D) E)
13.5 I only II only III only II and III I, II and III
Q8. What is the value of a + b + c+ d + e + f? A) B) C) D) E)
180 270 300 360 720
Q9. In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD? A) B) C) D)
3 15/4 5 16/3
E) 20/3
Q10.
What is the perimeter of the figure above? A) B) C) D) E)
24 25 28 30 36
Working towards Perfection: Practice Problems
Q1. In the figure above, sideAC of ABC is on line . What is x in terms of k? A) B) C) D) E)
60 – k k 60 + k 120 – k 120 – 2k
Q2. In triangle XYZ above, XW = 2, WZ = 8, and XY = 6. What is the area of triangle WYZ? A) B) C) D) E)
6 12 18 24 30
Q3. The figure above is a right triangle. What is the value of 25 +x2 ? A) 32 B) 34 C) 39 D) 50 E) 64
Q4. In the figure above, a < 40 and b = c + 1. If c is an integer, what is the least possible value of b?
A) 30 B) 39 C) D) 50 61 E) 71
Q5. Which of the following inequalities is true about the lengthsa and b of the sides of the triangle above?
A) 0 ≤ (a + b)2 < 20
B) C) D) E)
20 ≤ (a + b)2 < 40 40 ≤ (a + b)2 < 100 100 ≤ (a + b)2 < 400 400 ≤ (a + b)2
Q6. Triangles ABC and ACD in the figure above are equilateral. What is the ratio ofBD to AC?
A) B) C) D) E)
√2 to 1 √3 to 1 √2 to 2 √3 to 2 √3 to √2
Q7. If the area of an equilateral triangle isx square meters and the perimeter isx meters, then what is the length of one side of the triangle in meters? A) B) C) D) E)
6 8 4 √2 2√3
4√3
Q8. On a map Town G is 10 centimeters due east of TownH and 8 centimeters due south of Town J. Which of the following is closest to the straight-line distance, in centimeters, between Town H and Town J on the map? A) B) C) D) E)
6 13 18 20 24
Q9. In isosceles triangle PQR, if the measure of angle P is 80, which of the following could be the measure of angle R? I. II. III.
20 50 80
A) B) C) D) E)
I only III only I and II only II and III only I, II and III
Q10.
In the figure above, AE and
CD
are each perpendicular to CE . If x y , the length of AB is
4, and the length of BD is 8, what is the length of CE ? A)
3 2
B)
6 2
(approximately 4.24) (approximately 8.49)
C)
8 2
(approximately 11.31)
D)
10 2
(approximately 14.14)
E)
12 2
(approximately 16.97)
Playing four squares: Quadrilaterals A quadrilateral is a four-sided polygon, and a polygon is any closed figure made of line segments that intersect. Your primary concern is to know how to find quadrilaterals’ areas and perimeters. The measure of their perimeters is always the sum of their sides. The sum of the angle measures of a quadrilateral is always 360°.
Drawing parallels: Parallelograms
Most of the GRE quadrilaterals are parallelograms. Parallelograms have properties that are very useful for solving GRE problems: The opposite sides are parallel and equal in length.
The opposite angles are equal in measure to each other.
The measures of the adjacent angles add up to 180°, so they’re supplementary to each other.
The diagonals of a parallelogram bisect each other. In other words, they cross at
the midpoint of both diagonals. The area of any parallelogram is its base times its height A( = bh). You determine the height pretty much the same way you determine the height, or altitude, of a triangle. The difference is that you draw the perpendicular line from the base to the opposite side (instead of to the opposite angle, as in the case of a triangle).
You can use the Pythagorean theorem to help you find the height of a parallelogram. When you drop a perpendicular line from one corner to the base to create the height, the line becomes the leg of a right triangle. If the problem gives you the length of other sides of the triangle (or information you can use to determine the length), you can use the formula to find the length of the height.
Parallelograms come in various types:
A rectangle is a parallelogram with four right angles. Because rectangles are parallelograms, rectangles have all the properties of parallelograms. UseA = bh to find the area of a rectangle. The cool thing about rectangles, though, is that the height, or altitude, is the same as one of its sides.
A square is a rectangle with four equal sides. It has four right angles, and its sides all have the same length. Because a square has four equal sides, you can easily find its area if you know the length of only one side. The area of a square can be expressed as A = s2 where s is the length of a side. The perimeter of a square is 4 s. Here’s a neat trick for finding the area of a square if the only measurement you know is the length of
the diagonal. You can say A = d2 ⁄ 2, where the diagonal is d. Remember that the diagonal of a square is the hypotenuse of an isosceles right triangle, and right triangles have some special formulas. This shortcut is just a way of using the Pythagorean theorem in reverse!
A rhombus is a type of parallelogram. All four sides of a rhombus are equal in length, like
a square, but a rhombus doesn’t necessarily have four right angles. You can find the area of a rhombus by multiplying the lengths of the two diagonals (the straight lines that join opposite angles of the parallelogram, designated as d) and then dividing by 2, or A = 1⁄2d1d2.
Raising the roof: Trapezoids
A trapezoid is a quadrilateral with two parallel sides and two nonparallel sides. The parallel sides are called the bases, and the other two sides are called the legs. Finding the area of a trapezoid is a bit tricky, but it can be done as long as you know the length of both bases and the height, or altitude. To find the area, you take the average of the two bases and multiply by the height or altitude. Thus,A = 1⁄2(b1 + b2) × h. Let’s try some questions: Q1. A, B and C are three rectangles. The length and width of rectangle A are 10 percent greater and 10 percent less, respectively than the length and width of rectangle C. The length and width of rectangle B are 20 percent greater and 20 percent less, respectively, than the length and width of rectangle C. Column A The area of rectangle A
Column B The area of rectangle B
A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given Q2. The perimeter of a square equals the perimeter of a rectangle that is not a square. Column A Column B The area of the square The area of the rectangle A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given
Q3. In the figure above, if the area of triangleCAF is equal to the area of rectangleCDEF, what is the length of segmentAD? A) B) C) D) E)
7/2 5 7 15/2 15
Q4. In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square? A) B) C) D) E)
8√2 12
12√2 16 18
Going Round, round: Circles A circle, by technical definition, is a set of points in a plane that are at a fixed distance from a given point. That point is called the center. A circle is best drawn with the aid of a compass, but on the GREyou’ll just have your pencil.
Ring measurements: Radius, diameter, and circumference
Almost any GRE problem regarding circles requires you to know about radius, diameter, and circumference. The radius of a circle is the distance from the center of the circle to any point on the circle. Think of it as aray going out from the center to the edge of the circle. The diameter of a circle is the length of a line that goes from one side of the circle to the other and passes through the center. The diameter is twice the length of the radius, and it’s the longest possible distance across the circle. The circumference of a circle is the distance around the circle. You can think of the circumference as the perime ter of the circle, although this isn’t quite true. It’s really more
technically accurate to say you’re trying to find the perimeter of a regular polygon with an infinite number of sides as it gets rounder and rounder. Rather than taking the time to
figure out how many sides add up to infinity, just use this formula: C = 2πr or C = πd Area of the circle is given by: A = πr2
All about arcs
You should have a basic understanding of the following terms so you aren’t running in circles on the GRE math section: o An arc of a circle is a portion along the edge of the circle. Because it runs along the circumference, an arc is actually a part of the circle. o
A central angle of a circle is an angle that’s formed by two radii; it’s called a central angle because its vertex is the center of the circle. The measurement of the central angle is the same as that of the arc formed by the endpoints of its radii. So a 90° central angle intercepts one-quarter of the circle, or a 90° arc.
o
Length of the arc, L = ( /360)*2πr
Let’s try some questions: Q1. If three different circles are drawn on a piece of paper, at most how many points can be common to all three? A) None B) One C) Two D) Three E) Six
Q2. The circumference of the circle with center O shownabove is 2 shaded region?
What is the area of the
A) π/2 B) /4 C) 1
D) 1⁄2 E) 1⁄4
Q3. In the figure above, the smaller circles each have radius 3. They are tangent to the larger circle at points A and C, and are tangent to each other at point B, which is the center of the larger circle. What is the perimeter of the shaded region?
A) 6π B) 8 C) 9 D) 12 E) 15
Q4. In the figure above, triangle ABC is inscribed in the circle with center O and diameter AC. If AB = AO, what is the degree measure of ABO? A) 15 B) 30 C) 45 D) 60 E) 90
Q5. The figure above consists of two circles that have the same center. If the shaded area is
64
square inches and the smaller circle has a radius of 6 inches, what is the radius, in
inches, of the larger circle?
Working towards perfection:
Q1. Two circular road signs are to be painted yellow. If the radius of the larger sign is twice that of the smaller sign, how many times more paint is needed to paint the larger sign (assuming that a given amount of paint covers the same area on both signs)? A) 2 B) 3
C) π D) 4
E) 3
/2
Q2. Two circles share a center at point C, as shown. Segment AC is broken up into two shorter segments, AB and BC, with dimensions shown. What is the ratio of the area of the large circle to the area of the small circle? A) 25/4 B) 5/2 C) 3/2 D) 2/5 E) 4/25
Q3. The length of minor arc AB is twice the length of minor arc BC and the length of minor arc AC is three times the length of minor arc AB. What is the measure of angle BCA? A) 20 B) 40 C) 60 D) 80 E) 120
Solid Foundations: 3D Figures Three-dimensional geometry, or solid geometry, puts some depth to plane geometrical figures. Three-dimensional geometry is about as simple as plane geometry, and you can apply many of the same strategies. You’ll most likely be asked no more than a handfulof solid geometry questions on the GRE, and they’ll likely concern only rectangular solids and cylinders.
Chipping off the old block: Rectangular solids You make a rectangular solid by taking a simple rectangle and adding depth. Good examples of a rectangular solids are bricks, cigar boxes, or boxes of your favorite cereal. A rectangular solid has three dimensions: length, height, and width. You really only need to worry about two basic measurements of rectangular solids on the GRE: total surface area and volume.
Finding volume The volume (V) of a rectangular solid is a measure of how much space it occupies, or to put it in terms everyone can appreciate, how much cereal your cereal box holds. You measure the volume of an object in cubic units. The formula for the volume of a rectangular solid is simply its length (a) × width (b) × height (c). V = abc
Another way of saying this is that the volume is equal to the base times the height (V = Bh), where B is the area of the base.
Determining surface area You can find the surface area(SA) of a rectangular solid by simply figuring out the areas of all six sides of the object and adding them together. First you find the area of the length(a) times height (c), then the area of length times width (b), and finally width times height. Now multiply each of these three area measurements times two (after you find the area of one side, you know that the opposite side has the same measurement). The formula for the surface area of a rectangular solid is SA = 2ab + 2bc + 2ac You can visualize the surface area of a rectangular solid, or any solid figure for that matter,
by mentally flattening out all of the sides and putting them next to each other. It’s sort of like taking apart a cardboard box to get it ready for recycling, only now you get to measure it. Lucky you!
Sipping from soda cans and other cylinders A cylinder is a circle that grows straight up into the third dimension to become the shape of a can of soda. The on bases a cylinder arecircular two congruent different planes. The cylinders you see the of GRE are right cylinders,circles whichonmeans that the line segments that connect the two bases are perpendicular to the bases. All the corresponding points on the circles are joined together by line segments. The line segment connecting the center of one circle to the center of the opposite circle is called the axis.
A right circular cylinder has the same measurements as a circle. That is, a right circular cylinder has a radius, diameter, and circumference. In addition, a cylinder has a third dimension, its height, or altitude. To get the volume of a right circular cylinder, first take the area of the base (a circle), which is πr2, and multiply by the height(h) of the cylinder. Here’s the formula: V = πr2h If you want to find the total surface area of a right circular cylinder, you have to add up the areas of all the surfaces. Imagine taking a soda can, cutting off the top and bottom sections, and then slicing it down one side. You then spread out the various parts of the can. If you measure each one of these sections, you get the total surface area.
When you measure the surface area of a right circular cylinder, don’t forget to include the top and bottom of the can in your calculation. Here’s the formula for the total surface area (TSA) of a right circular cylinder — the diameter (d) is 2 times the radius(r):
TSA = πdh + 2πr2 Let’s Try Some Questions: Q1. How many cubical blocks, each with edges of length 4 centimeters, are needed to fill a rectangular box that has inside dimensions 20 centimeters by 24 centimeters by 32 centimeters? A) 38 B) 96 C) 192 D) 240 E) 384
Q2. The three–dimensional figure above has two parallel bases and 18 edges. Line segments are to be drawn connecting vertex V with each of the other 11 vertices in the figure. How many of these segments will not lie on an edge of the figure? A) Two B) Three C) Four D) Seven E) Nine Q3. In a windowless, cube-shaped storage room, the ceiling and 4 walls, including a door, are completely painted. The floor is not painted. If the painted area is equal to 80 square meters, what is the volume of the room, in cubic meters? A) 16 B) 20 C) 64 D) 256 E) 400 Q4. What is the total number of right angles formed by the edges of a cube? A) 36 B) 24 C) 20 D) 16 E) 12
Looking for location: Co-ordinate Geometry Co-ordinate geometry combines the study of algebra and planes with three-dimensional geometry. You can expect to encounter coordinate geometry questions on roughly ten percent of the problems on the GRE. If you’re not particularly savvy about coordinate geometry, it won’t significantly affect your GRE math score.
Taking Flight: The Coordinate Plane
The coordinate plane doesn’t have wings, but it does have points that spread out infinitely. You may not have encountered the coordinate plane in a while (it isn’t something most people choose to deal with in everyday life), so take just a minute to refresh your memory about a few relevant terms that may pop up on the GRE . Although you won’t be asked to define these terms, knowing what they mean is absolutely essential. Here are some coordinate geometry terms that show up from time to time on the GRE:
Coordinate plane: The coordinate, or Cartesian, plane is a perfectly flat surface that contains a system in which points can be identified by their position using an
ordered pair of numbers. This pair of numbers represents the points’ distance from an srcin on perpendicular axes. The coordinate of any particular point is the set of numbers that identifies the location of the point, such as (3, 4) or (x, y).
x -axis:
The x-axis is the horizontal axis (number line) on a coordinate plane in which values or numbers start at the srcin, which has a value of 0. Numbers increase in value to the right of the srcin and decrease in value to the left. Thex
value of a point’s coordinate is listed first.
y -axis:
The y-axis is the vertical axis (number line) on a coordinate plane in which values or numbers start at the srcin, which has a value of 0. Numbers increase in value going up from the srcin and decrease in value going down. They value of a
point’s coordinate is listed second.
Origin: The srcin is the point (0, 0) on the coordinate plain. It’s where thex- and yaxes intersect.
Ordered pair: Also known as a coordinate pair, this is the set of two numbers that shows the distance of a point from the srcin. The horizontal (x) coordinate is
always listed first, and the vertical coordinate(y) is listed second. A ordered pair will always be (x, y).
Ordinate: Ordinate is another way of referring to they-coordinate in an ordered pair. Used very rarely by the GRE.
x -intercept:
y -intercept:
Slope: Slope measures how steep a line is and is commonly referred to asthe rise over the run. Slope = (y2 – y1)/(x2 – x1)
The value of x where a line, curve, or some other function crosses the x-axis. The value of y is 0 at the x-intercept. The x-intercept is often thesolution or root of an equation. The value of y where a line, curve, or some other function crosses the y-axis. The value of x is 0 at the y-intercept.
On all fours: Quadrants The intersection of the x- and y-axes forms four quadrants on the coordinate plane, which
just so happen to be named Quadrants I, II, III, and IV . Here’s what you can assume about points based on the quadrants they’re in:
All points in Quadrant I have a positivex value and a positivey value.
All points in Quadrant II have a negativex value and a positivey value. All points in Quadrant III have a negativex value and a negativey value.
All points in Quadrant IV have a positivex value and a negativey value.
All points along the x-axis have a y value of 0.
All points along they-axis have an x value of 0.
Let’s try some questions: Q1. In the xy-plane, line k is a line that does not pass through the srcin. Which of the following statements individually provide(s) sufficient additional information to determine whether the slope of linek is negative? Indicate all such statements. A) The x-intercept of line k is twice they-intercept of line k B) The product of the x-intercept and they-intercept of line k is positive C) Line k passes through the points (a, b) and (r, s), where (a – r)(b – s) < 0. Q2. In the xy-plane, the point with coordiantes (-6, -7) is the center of circleC. The point with coordinates (-6, 5) lies inside C, and the point with coordiantes (8, -7) lies outsideC. If m is the radius of C and m is an integer, what is the value ofm? Q3.
Line k goes through (1, 1) and (5, 2). Line m is perpendicular tok. Column A The slope of line k
Column B The slope of line m
A) Quantity A is greater B) Quantity B is greater C) The two quantities are equal D) The relationship cannot be determined from the information given.
Q4. In the figure above, point P(-√3, 1) and Q(s, t) lie on the circle with center O. What is the value of s? A) B) C) D)
√3 1
√2 2
E) 5 Q5. Line l is defined by the equation y– 5x = 4 and line w is defined by the equation 10y + 2x + 20 = 0. If line k does not intersect line l, what is the degree measure of the angle formed by line k and line w? A) B) C) D) E)
0 30 60 90 Cannot be determined from the information given
Q6. A certain square is to be drawn on a coordinate plane. One of the vertices must be on the srcin and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn? A) B) C) D) E)
4 6 8 10 12
Q7. In the rectangular coordinate system, a line passes through the points (0, 5) and (7, 0). Which of the following points must the line also pass through? A) B) C) D) E)
(-14, 10) (-7, 5) (12, -4) (14, -5) (21, -9)
Working towards Perfection: Practice Problems
Q1. Point P is the point with the greatesty-coordinate on the semicircle shown above. What is the x-coordinate of pointQ? A) B) C) D) E)
-3.5 -3 -2.5 -2 -1.5
Q2. In the xy -plane, an equation of line isy = 3x −1. If line m is the reflection of line in they -axis, what is an equation of linem? A) B) C) D) E)
y = -3x – 1 y = -3x + 1 y = 3x + 1 y = -x/3 – 1 y = -x/3 + 1
Q3. In the figure above, if line l has a slope of -2, what is the y-intercept of line l? A) 7 B) 8 C) 9 D) 10
E) 12
Q4. If points (0, -3), (6, 0) and k( , 10) all lie on the same line, what is the value ofk? A) B) C) D) E)
2 8 14 22 26
Q5. In the xy-coordinate plane, the points (a, b) and (a + k, b - 3) are on the line defined by y = 2x – 5. What is the value of k? A) B) C) D) E)
-5/2 -5/3 -3/2 -2/3 -2/5
Q6. In the xy-plane, the point (-2, -3) is the center of a circle. The point (-2, 1) lies inside the circle and the point (4, -3) lies outside the circle. If the radius r of the circle is an integer, then r = A) B) C) D) E)
6 5 4 3 2
Q7. A certain quantity is measured on two different scales, theR-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on theRscale corresponds to a measurement of 100 on theS-scale? A) B) C) D) E)
20 36 48 60 84
