Tài liệu Master the Gre 2010 - Part 32 pdf

Or, it might ask you to determine the equation of a line, or just the line’s slope m or y-intercept b, given the coordinates of two points on the line.. Graphing a Line on the XY-PlaneYo

y-coordinate is its vertical position on the plane You denote the coordinates of a point

with (x,y), where x is the point’s x-coordinate and y is the point’s y-coordinate.

The center of the coordinate plane—the intersection of the x- and y-axes—is called the

origin The coordinates of the origin are (0,0) Any point along the x-axis has a

y-coordinate of 0 (x,0), and any point along the y-axis has an x-coordinate of 0 (0, y).

The coordinate signs (positive or negative) of points lying in the four Quadrants I–IV

in this next figure are as follows:

Notice that we’ve plotted three different points on this plane Each point has its own

unique coordinates (Before you continue, make sure you understand why each point

is identified by two coordinates.)

Defining a Line on the XY-Plane

You can define any line on the coordinate plane by the equation:

y 5 mx 1 b

In this equation:

• The variable m is the slope of the line.

• The variable b is the line’s y-intercept (where the line crosses the y-axis).

• The variables x and y are the coordinates of any point on the line Any (x,y) pair

defining a point on the line can substitute for the variables x and y.

Determining a line’s slope is often crucial to solving GRE coordinate geometry

problems Think of the slope of a line as a fraction in which the numerator indicates

the vertical change from one point to another on the line (moving left to right)

corresponding to a given horizontal change, which the fraction’s denominator

indi-cates The common term used for this fraction is “rise over run.”

You can determine the slope of a line from any two pairs of (x,y) coordinates In

general, if (x1,y1) and (x2,y2) lie on the same line, calculate the line’s slope as follows

slope ~m! 5 y22 y1

x22 x1 or

y12 y2

x12 x2

In applying the preceding formula, be sure to subtract corresponding values For

example, a careless test taker calculating the slope might subtract y1 from y2 but

subtract x2from x1 Also, be sure to calculate “rise-over-run,” and not “run-over-rise”—

another relatively common error

As another example, here are two ways to calculate the slope of the line defined by the two points P(2,1) and Q(23,4):

slope ~m! 5 4 2 1

23 2 25

3 25

slope ~m! 5 1 2 4

2 2 ~23!5

23 5

A GRE question might ask you to identify the slope of a line defined by a given

equation, in which case you simply put the equation in the standard form y 5 mx 1 b, then identify the m-term Or, it might ask you to determine the equation of a line, or just the line’s slope (m) or y-intercept (b), given the coordinates of two points on the

line

29 On the xy-plane, at what point along the vertical axis (the y-axis) does the

line passing through points (5, 22) and (3,4) intersect that axis?

(A) 28

(B) 25

2

(C) 3 (D) 7 (E) 13

The correct answer is (E) The question asks for the line’s y-intercept (the

value of b in the general equation y 5 mx 1 b) First, determine the line’s slope:

slope m 5 y22 y1

x22 x15

4 2 ~22!

3 2 5 5

6

225 23

In the general equation (y 5 mx 1 b), m 5 23 To find the value of b, substitute either (x,y) value pair for x and y, then solve for b Substituting the (x,y) pair (3,4):

y 5 23x 1 b

4 5 23~3! 1 b

4 5 29 1 b

ALERT!

On the xy-plane, a line’s slope

is its “rise over run”—the

vertical distance between two

points divided by the

horizontal distance between

the same two points When

finding a slope, be careful not

to calculate “run over rise”

instead!

To determine the point at which two nonparallel lines intersect on the coordinate

plane, first determine the equation for each line Then, solve for x and y by either

substitution or addition-subtraction

30 In the standard xy-coordinate plane, the xy-pairs (0,2) and (2,0) define a

line, and the xy-pairs (22,21) and (2,1) define another line At which of

the following points do the two lines intersect?

(A)S4

3, 2

2, 4

2, 3 2D

(D)S3

4, 2

2

4, 2

2

3D

The correct answer is (A) Formulate the equation for each line by

determining slope (m), then y-intercept (b) For the pairs (0,2) and (2,0):

y 5S0 2 2

2 2 0Dx 1 b ~slope 5 21!

0 5 22 1 b

2 5 b

The equation for the line is y 5 2x 1 2 For the pairs (22, 21) and (2,1):

y 5S1 2 ~21!2 2 ~22!Dx 1 bSslope 512D

1 51

2~2! 1 b

0 5 b

The equation for the line is y 51

2x To find the point of intersection, solve for

x and y by substitution For example:

1

2x 5 2x 1 2

3

2x 5 2

x 5 4

3

y 5 2

3

The point of intersection is defined by the coordinate pairS4

3, 2

3D

Graphing a Line on the XY-Plane

You can graph a line on the coordinate plane if you know the coordinates of any two points on the line Just plot the two points and then draw a line connecting them You can also graph a line from one point on the line, if you know either the line’s slope or

its y-intercept.

A GRE question might ask you to recognize the value of a line’s slope (m) based on a

graph of the line If the graph identifies the precise coordinates of two points, you can determine the line’s precise slope and the entire equation of the line Even without any precise coordinates, you can still estimate the line’s slope based on its appearance

Lines That Slope Upward from Left to Right:

• A line sloping upward from left to right has a positive slope (m).

• A line with a slope of 1 slopes upward from left to right at a 45° angle in relation

to the x-axis.

• A line with a fractional slope between 0 and 1 slopes upward from left to right but

at less than a 45° angle in relation to the x-axis.

• A line with a slope greater than 1 slopes upward from left to right at more than a

45° angle in relation to the x-axis.

Lines That Slope Downward from Left to Right:

• A line sloping downward from left to right has a negative slope (m).

• A line with a slope of 21 slopes downward from left to right at a 45° angle in

relation to the x-axis.

• A line with a fractional slope between 0 and 21 slopes downward from left to

right but at less than a 45° angle in relation to the x-axis.

• A line with a slope less than 21 (for example, 22) slopes downward from left to

right at more than a 45° angle in relation to the x-axis.

Horizontal and Vertical Lines:

• A horizontal line has a slope of zero (0) (m 5 0, and mx 5 0).

• A vertical line has either an undefined or an indeterminate slope (the fraction’s

denominator is zero (0))

TIP

Parallel lines have the same

slope (the same m-term in the

general equation) The slope

of a line perpendicular to another is the negative reciprocal of the other line’s slope (The product of the two slopes is 21.)

31. P

Referring to the xy-plane above, which of the following could be the

equation of line P?

(A) y 52

5x 2

5

2 (B) y 5 2

5

2x 1

5

2 (C) y 5

5

2x 2

5 2

(D) y 52

5x 1

2

5 (E) y 5 2

5

2x 2

5 2

The correct answer is (E) Notice that line P slopes downward from left

to right at an angle greater than 45° Thus, the line’s slope (m in the equation y 5 mx 1 b) , 2 1 Also notice that line P crosses the y-axis at a negative y-value (that is, below the x-axis) That is, the line’s y-intercept (b

in the equation y 5 mx 1 b) is negative Only choice (E) provides an

equation that meets both conditions

Midpoint and Distance Formulas

To be ready for GRE coordinate geometry, you’ll need to know midpoint and distance formulas To find the coordinates of the midpoint of a line segment, simply average

the two endpoints’ x-values and y-values:

x M5x11 x2

2 and y M5

y11 y2

2

For example, the midpoint between (23,1) and (2,4) 5S23 1 2

2 ,

1 1 4

2 DorS21

2,

5

2D

A GRE question might simply ask you to find the midpoint between two given points,

or it might provide the midpoint and one endpoint and then ask you to determine the other point

32 In the standard xy-coordinate plane, the point M(21,3) is the midpoint of a

line segment whose endpoints are A(2,24) and B What are the

xy-coordinates of point B?

(A) (21,22) (B) (23,8) (C) (8,24)

The correct answer is (E) Apply the midpoint formula to find the x-coordinate

of point B:

21 5x 1 2

2

22 5 x 1 2

24 5 x

Apply the midpoint formula to find the y-coordinate of point B:

3 5y 2 4

2

6 5 y 2 4

10 5 y

To find the distance between two points that have the same x-coordinate (or

y-coor-dinate), simply compute the difference between the two y-values (or x-values) But if

the line segment is neither vertical nor horizontal, you’ll need to apply the distance

formula, which is actually the Pythagorean theorem in thin disguise (it measures the

length of a right triangle’s hypotenuse):

d 5=~x12 x2!21 ~y12 y2!2

For example, the distance between (23,1) and (2,4) 5 =~23 2 2!21 ~1 2 4!25

=25 1 9 5=34

A GRE question might ask for the distance between two defined points, as in the

example above Or it might provide the distance, and then ask for the value of a

missing coordinate—in which case you solve for the missing x-value or y-value in the

formula

Figures in Two Dimensions

Up to this point in the chapter, the coordinate geometry tasks you’ve learned to

perform have all involved points and lines (line segments) only In this section, you’ll

explore coordinate-geometry problems involving two-dimensional geometric figures,

especially triangles and circles

Triangles and the Coordinate Plane

On the GRE, a question might ask you to find the perimeter or area of a triangle

defined by three particular points As you know, either calculation requires that you

know certain information about the lengths of the triangle’s sides Apply the distance

formula (or the standard form of the Pythagorean theorem) to solve these problems

33 On the xy-plane, what is the perimeter of a triangle with vertices at points

A(21,23), B(3,2), and C(3,23)?

(A) 12 (B) 10 1 2=3

(C) 7 1 5=2

(D) 15 (E 9 1=41

The correct answer is (E) The figure below shows the triangle on the

coor-dinate plane:

B

C A

AC 5 4 and BC 5 5 Calculate AB (the triangle’s hypotenuse) by the distance formula or, since the triangle is right, by the standard form of the Pythagorean theorem: (AB)2 5 42 1 52; (AB)2 5 41; AB 5 =41 The triangle’s perimeter 5 4 1 5 1=41 5 9 1=41

Note that, since the triangle is right, had the preceding question asked for the triangle’s area instead of perimeter, all you’d need to know are the lengths of the two legs (AC and BC) The area isS1

2D~4!~5!5 10

To complicate these questions, the test makers might provide vertices that do not connect to form a right triangle Answering this type of question requires the extra step of finding the triangle’s altitude Or they might provide only two points, then require that you construct a triangle to meet certain conditions

34 On the xy-plane, the xy-coordinate pairs (26,2) and (214,24) define one

line, and the xy-coordinate pairs (212,1) and (23,211) define another line.

What is the unit length of the longest side of a triangle formed by the

y-axis and these two lines?

(A) 15

(B) 17.5

(C) 19

(D) 21.5

(E) 23

The correct answer is (D) For each line, formulate its equation by

deter-mining slope (m), then y-intercept (b).

For the Pairs (26,2) and

(214,24)

For the Pairs (212,1) and (23,211)

y 5 6

8x 1 bSslope 53

4D

2 5 3

4~26! 1 b

2 5 241

21 b

2 1 41

2 5 b

61

2 5 b

y 5 212

9 x 1 bSslope 5 24

3D

1 5 24

3~212! 1 b

1 5 48

3 1 b

1 2 16 5 b

215 5 b

The two y-intercepts are 61

2and 215 Thus the length of the triangle’s side along

the y-axis is 21.5 But is this the longest side? Yes Notice that the slopes of the

other two lines (l1and l2in the next figure) are negative reciprocals of each other:

S3

4DS24

3D5 21 This means that they’re perpendicular, forming the two legs of

a right triangle in which the y-axis is the hypotenuse (the longest side).

If the preceding question had instead asked for the point at which the two lines intersect, you would answer the question formulating the equations for both lines,

then solving for x and y with this system of two equations in two variables.

Circles and the Coordinate Plane

A GRE question might ask you to find the circumference or area of a circle defined by

a center and one point along its circumference As you know, either calculation requires that you know the circle’s radius Apply the distance formula (or the standard form of the Pythagorean theorem) to find the radius and to answer the question

35 On the xy-plane, a circle has center (2,21), and the point (23,3) lies along

the circle’s circumference What is the square-unit area of the circle?

(A) 36p

(B) 81p

2

(C) 41p (D) 48p (E) 57p The correct answer is (C) The circle’s radius is the distance between its center

(2,21) and any point along its circumference, including (23,3) Hence, you can

find r by applying the distance formula:

=~23 2 2!21 ~3 2 ~21!!25=25 1 16 5=41 The area of the circle 5 p~=41!25 41p

In any geometry problem involving right triangles, keep your eyes open for the Pythagorean triplet in which you’ll see the correct ratio, but the ratio is between the wrong two sides For instance, in the preceding problem, the lengths of the two legs of

a triangle whose hypotenuse is the circle’s radius are 4 and 5 But the triangle does

not conform to the 3:4:5 Pythagorean side triplet Instead, the ratio is 4:5:=41