Tài liệu Master the Gre 2010 - Part 29 docx

After a brief review of the properties of any triangle, we’ll focus on right triangles which include one right, or 90°, angle, isosceles triangles in which two sides are congruent, and e

x˚

40˚

y˚

The figure above shows three intersecting lines What is the value of

x 1 y ?

(A) 50

(B) 80

(C) 130

(D) 140

(E) 150

The correct answer is (D) The angle vertical to the one indicated as being 40°

must also measure 40° That 40° angle, together with the angles whose measures

are x° and y°, combine to form a straight (180°) line In other words, 40 1 x 1 y 5

180 Thus, x 1 y 5 140.

A slightly tougher “wheel-spoke” question might focus on overlapping angles and

require you to apply rule 1 (about vertical angles) to determine the amount of the

overlap Look at this “wheel-spoke” figure:

A GRE question about this figure might test your ability to recognize one of the

following relationships:

NOTE

Degree measures of the individual central angles of a wheel-spoke figure total 360°, which is the number of degrees in a circle You’ll explore this concept again later in the chapter.

x° 1 y° 2 z° 5 180°

x° 1 y° exceeds 180° by the amount of the overlap, which equals z°, the angle

vertical to the overlapping angle

x° 1 y° 1 v° 1 w° 5 360°

The sum of the measures of all angles, excluding z°, is 360°; z is excluded because it is already accounted for by the overlap of x and y.

y° 2 w° 5 z°

w° equals its vertical angle, so y 2 w equals the portion of y vertical to

angle z.

Parallel Lines and Transversals

GRE problems involving parallel lines also involve at least one transversal, which is a line that intersects each of two (or more) lines Look at this next figure, in which

The upper-left “cluster” of angles (numbered 1, 2, 3, and 4) matches each of the three other clusters In other words:

If you know the size of just one angle, you can determine the size of all 16 angles

(A) 75°

(B) 85°

(C) 95°

(D) 105°

(E) 115°

The correct answer is (D) The angle “cluster” where lines P and R intersect

must measure 105°

TRIANGLES

The triangle (a three-sided polygon) is the test makers’ favorite geometric figure.

You’ll need to understand triangles not only to solve “pure” triangle problems but also

to solve certain problems involving four-sided figures, three-dimensional figures, and

even circles After a brief review of the properties of any triangle, we’ll focus on right

triangles (which include one right, or 90°, angle), isosceles triangles (in which two

sides are congruent), and equilateral triangles (in which all sides and angles are

congruent)

Properties of All Triangles

Here are four properties that all triangles share:

Length of the sides: Each side is shorter than the sum of the lengths of the other

two sides (Otherwise, the triangle would collapse into a line.)

Angle measures: The measures of the three angles total 180°.

Angles and opposite sides: Comparative angle sizes correspond to the

com-parative lengths of the sides opposite those angles For example, a triangle’s

largest angle is opposite its longest side (The sides opposite two congruent angles

are also congruent.)

ALERT!

The ratio of angle sizes need not be identical to the ratio of lengths of sides For example, if

a certain triangle has angle measures of 30°, 60°, and 90°, the ratio of the angles is 1:2:3 But this doesn’t mean that the ratio of the opposite sides is also 1:2:3 It’s not!

Area: The area of any triangle is equal to one-half the product of its base and its

calculate area

Right Triangles and the Pythagorean Theorem

In a right triangle, one angle measures 90° (and, of course, each of the other two

angles measures less than 90°) The Pythagorean theorem expresses the relationship

and c is the hypotenuse—the longest side, opposite the right angle:

For any right triangle, if you know the length of two sides, you can determine the length of the third side by applying the theorem

If the two shortest sides (the legs) of a right triangle are 2 and 3 units long, then the

If a right triangle’s longest side (hypotenuse) is 10 units long and another side (one of

PYTHAGOREAN SIDE TRIPLETS

A Pythagorean side triplet is a specific ratio among the sides of a triangle that satisfies the Pythagorean theorem In each of the following triplets, the first two numbers represent the comparative lengths of the two legs, whereas the third—and greatest—number represents the comparative length of the hypotenuse (on the GRE, the first four triplets appear far more frequently than the last two):

Each triplet above is expressed as a ratio because it represents a proportion among

the triangle’s sides All right triangles with sides having the same proportion, or ratio, have the same shape For example, a right triangle with sides of 5, 12, and 13 is

smaller but exactly the same shape (proportion) as a triangle with sides of 15, 36,

and 39

3 Two boats leave the same dock at the same time, one traveling due east at

10 miles per hour and the other due north at 24 miles per hour How

many miles apart are the boats after 3 hours?

(A) 68

(B) 72

(C) 78

(D) 98

(E) 110

The correct answer is (C) The distance between the two boats after 3 hours

forms the hypotenuse of a triangle in which the legs are the two boats’ respective

paths The ratio of one leg to the other is 10:24, or 5:12, so you know you’re

dealing with a 5:12:13 triangle The slower boat traveled 30 miles (10 mph 3 3

hours) Thirty corresponds to the number 5 in the 5:12:13 ratio, so the multiple is

6 (5 3 6 5 30) 5:12:13 5 30:72:78

PYTHAGOREAN ANGLE TRIPLETS

In two (and only two) of the unique triangles identified in the preceding section as

Pythagorean side triplets, all degree measures are integers:

and 90°

and 90°

If you know that the triangle is a right triangle (one angle measures 90°) and that one

of the other angles is 45°, then given the length of any side, you can determine the

unknown lengths For example:

10

TIP

To save valuable time on GRE right-triangle problems, learn

to recognize given numbers (lengths of triangle sides) as multiples of Pythagorean triplets.

Similarly, if you know that the triangle is a right triangle (one angle measures 90°) and that one of the other angles is either 30° or 60°, then given the length of any side you can determine the unknown lengths For example:

units long (3 3 2)

4 In the figure below, AC is 5 units long, m∠ABD 5 45°, and m∠DAC 5 60° How many units long is BD ?

60˚

5

45˚

A

(A)7

2

(D)3=3

7 2

The correct answer is (C) To find the length of BD, you first need to

find AD Notice that DADC is a 30°-60°-90° triangle The ratio among its

5

Isosceles Triangles

An isosceles triangle has the following two special properties:

Two of the sides are congruent (equal in length)

The two angles opposite the two congruent sides are congruent (equal in size, or degree measure)

If you know any two angle measures of a triangle, you can determine whether the

triangle is isosceles

5.

A

40º

70º

How many units long is AB ?

(A) 5

(B) 6

(D) 7

(C) 8

(E) 9

The correct answer is (B) Since m∠A and m∠B add up to 110°, m∠C 5 70°

(70 1 110 5 180), and you know the triangle is isosceles What’s more, since

long

A line bisecting the angle connecting the two congruent sides divides an isosceles

triangle into two congruent right triangles So if you know the lengths of all three

sides of an isosceles triangle, you can determine the area of these two right triangles

by applying the Pythagorean theorem, as in the next example

6 Two sides of a triangle are each 8 units long, and the third side is 6 units

long What is the area of the triangle, expressed in square units?

(A) 14

(B) 12=3

(C) 18

(D) 22

(E) 3=55

The correct answer is (E) Bisect the angle connecting the two congruent sides

(as in DABC in the next figure) The bisecting line is the triangle’s height (h), and

the triangle’s base is 6 units long

B A

You can determine the triangle’s height (h) by applying the Pythagorean

theorem:

h25 55

h 5=55

A triangle’s area is half the product of its base and height Thus, the area of

Equilateral Triangles

An equilateral triangle has the following three properties:

All three sides are congruent (equal in length)

The measure of each angle is 60°

2=3

Any line bisecting one of the 60° angles divides an equilateral triangle into two right

triangles, as shown in the right-hand triangle in the next figure (Remember that Pythagorean angle triplet?)

C

60º

60º 60º

right-hand triangle in the preceding figure) The area of this equilateral triangle is 1

A quadrilateral is a four-sided geometric figure The GRE emphasizes four specific

types of quadrilateral: the square, the rectangle, the parallelogram (including the

rhombus), and the trapezoid In this section, you’ll learn the unique properties of each

type, how to distinguish among them, and how the GRE test makers design questions

about them

Rectangles, Squares, and Parallelograms

Here are five characteristics that apply to all rectangles, squares, and parallelograms:

The sum of the measures of all four interior angles is 360°

Opposite sides are parallel

Opposite sides are congruent (equal in length)

Opposite angles are congruent (the same size, or equal in degree measure)

Adjacent angles are supplementary (their measures total 180°)

A rectangle is a special type of parallelogram in which all four angles are right angles

(90°) A square is a special type of rectangle in which all four sides are congruent

(equal in length) For the GRE, you should know how to determine the perimeter and

area of each of these three types of quadrilaterals Referring to the next three figures,

here are the formulas (l 5 length and w 5 width):

Rectangle

Perimeter 5 2l 1 2w

Area 5 l 3 w

Perimeter 5 4s [s 5 side]

Parallelogram

Perimeter 5 2l 1 2w Area 5 base (b) 3 altitude (a)

GRE questions involving squares come in many varieties For example, you might need to determine area, given the length of any side or either diagonal, or perimeter

Or, you might need to do just the opposite—find a length or perimeter given the area For example:

The area of a square with a perimeter of 8 is 4

The area of a square with a diagonal of 6 is 18

Or, you might need to determine a change in area resulting from a change in perimeter (or vice versa)