Veritas prep GMAT geometry

Consider the following figure from a problem in a geometry book:Note that angle CED is equal to angle AEF because of intersecting lines , and that angle CDE equals angle EAF because of p

Brian Galvin

Chris Kane

Geometry

Markus MobergChad Troutwine

David NewlandAshley Newman-OwensJodi Brandon

Nick Mason

Tom AhnDennis Anderson

ALL RIGHTS RESERVED Printed in the U.S.A.

Third Edition, Copyright © 2013 by Veritas Prep, LLC.

GMAT® is a registered trademark of the Graduate

Management Admissions Council, which is not affiliated with this book.

No part of this publication may be reproduced, stored in

a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of Veritas Prep, LLC.

All the materials within are the exclusive property of Veritas Prep, LLC © 2013.

This book is dedicated to Veritas Prep’s instructors, whose enthusiasm and experience have contributed mightily to our educational philosophy and our students’ success.

It is also dedicated to the teachers who inspired Veritas Prep’s instructors The lesson that follows was only made possible by a lifelong love of learning and of undertaking educational challenges; we have teachers around the world to thank for that

Finally and most importantly, this book is dedicated to our thousands of students, who have taught us more about teaching and learning than they will ever know And to you, the reader, thank you for adding yourself to that group

Personal Dedications

Veritas Prep is a community of educators, students, and support staff, and these books would not be possible without our cast of thousands We thank you all, but would like to specifically acknowledge the following people for their inspiration:

Bogdan Andriychenko (GMAT Club), Clay Christensen (Harvard Business School), Tom Cotner (Plymouth-Salem High School), David Cromwell (Yale School of Management), Henry Grubb (Fort Osage High School), Dana Jinaru (Beat the GMAT), Steven Levitt (University of Chicago), Walter Lewin (Massachusetts Institute of Technology), Lawrence Rudner (Graduate Management Admissions Council), Jeff Stanzler (University of Michigan), and Robert Weber (Kellogg School of Management)

TABLE OF CONTENTS

LESSON PREVIEW 7

SKILLBUILDER 13

LESSON 53

Geometry: Leveraging Assets .53

Geometry and the Veritas Prep Pyramid 54

SECTION 1: GEOMETRY STRATEGY 55

Leveraging Assets 57

GMAT Geometry Cheat Sheet 58

SECTION 2: TRIANGLES 60

Essential Properties of Triangles 60

Right Triangles 61

Isosceles Triangles 65

Equilateral Triangles 67

Similar Triangles 71

External Supplementary Angles 73

Triangles Summary 76

SECTION 3: QUADRILATERALS 77

Essential Properties of Quadrilaterals 77

Quadrilaterals and Triangles 79

Defining Properties of Quadrilaterals 81

Diagonals 83

Border Problems 85

Quadrilaterals Summary 87

SECTION 4: CIRCLES 88

Essential Properties of Circles 88

Basic Circle Properties 89

Circles and Inscribed Angles 91

Shapes Within Shapes 93

Unusual Circle Figures 97

Circles Summary 99

table of contents

SECTION 5: COORDINATE GEOMETRY 101

Essential Properties of Coordinate Geometry 101

Graphing Lines in the Coordinate Plane 103

Mapping Figures in the Coordinate Plane 105

Coordinate Geometry Summary 108

SECTION 6: 3-DIMENSIONAL fIGURES .109

Essential Properties of Common 3-D Shapes 109

Common 3-D Figures 111

Unusual 3-D Figures 112

Dimensional Figures Summary 115

SECTION 7: YOU OUGHTA KNOW 116

Third Side Rule 117

Greatest Distance Between Two Points Shortcut 119

More on Unique 3-D Shapes 121

Data Sufficiency and Geometry 123

Problems in Which Figures Are Not Drawn to Scale 131

Geometry as a Vehicle for Hard Algebra 133

HOMEWORK PROBLEMS 135

ANSWER KEY 191

REMEMBERING Skillbuilder

In order to test higher-level thinking skills, testmakers must have some underlying content from which to create problems On the GMAT, this content is primarily:

• Math curriculum through the early high school level, and

• Basic grammar skills through the elementary school level

To succeed on the GMAT you must have a thorough mastery of this content, but many students already have a relatively strong command of this material For each content area, we have identified all core skills that simply require refreshing and/or memorizing and have put them in

our Skillbuilder section By doing this:

1 Students who need to thoroughly review or relearn these core skills can do so at their own pace, and

2 Students who already have a solid command of the underlying content will not become disengaged because of a tedious review of material they’ve already mastered.

APPLYING Skills Meet Strategy

What makes the GMAT difficult is not so much the underlying skills and concepts, but rather the way those skills and concepts are tested On the GMAT, what you know is only as valuable as what you can do with that knowledge The Veritas Prep curriculum emphasizes learning through challenging problems so that you can:

1 Learn how to combine skills and strategies to effectively solve any GMAT problem,

2 Most effectively utilize the classroom time you spend with a true GMAT expert, and

3 Stay focused and engaged, even after a long day in the office.

CREATING Think Like the Testmaker

Creating is the top of the pyramid in Bloom’s Taxonomy When you have completely mastered the GMAT, you are able to Think Like the Testmaker You are on top of the pyramid looking down! You don’t just have good content knowledge and lots of practice with GMAT problems; you understand how a problem has been made, what makes it hard, and how to break it down When you Think Like the Testmaker you can:

1 Quickly recognize what the problem is actually asking,

2 Discover hidden information and manipulate it to make it useful,

3 Recognize and see through trap answers, and

4 Create your own plan of attack for any problem.

As you learned in the Foundations of GMAT Logic lesson, the educational philosophy at

Veritas Prep is based on the multi-tiered Bloom’s Taxonomy of Educational Objectives,

which classifies different orders of thinking in terms of understanding and complexity

To achieve a high score on the GMAT, it is essential that you understand the test from

the top of the pyramid On the pages that follow, you will learn specifically how to

achieve that goal and how this lesson in particular relates to the Veritas Prep Pyramid

PREVIEW

How This Book Is Structured

Our Curriculum Is Designed to Maximize Your Time

The Veritas Prep Teaching Philosophy: Learning by Doing

Business schools have long featured the Case Method of education, providing students with real-world problems to solve by applying the frameworks they have studied The

Veritas Prep Learning by Doing method is similar In class, you will spend your time

applying skills and concepts to challenging GMAT problems, at the same time reviewing and better understanding core skills while focusing your attention on application and strategy The Case Method in business school maximizes student engagement and develops higher-order thinking skills, because students must apply and create, not just

remember Similarly, the Learning by Doing philosophy maximizes the value of your

study time, forcing you to engage with difficult questions and develop pyramid reasoning ability

top-of-the-An important note on Learning by Doing: In business school, your goal with a

business case is not to simply master the details of a particular company’s historical situation, but rather to develop broader understanding of how to apply frameworks

to real situations In this course, you should be certain to reflect on each question not simply through that narrow lens (Did you answer correctly? What key word made the difference?), but rather as an example of larger GMAT strategy (How could the exam bait you with a similar trap? How deeply do you need to understand the content to solve this genre of problem more efficiently?)

How This Book Is Structured

As you learned in the Foundations of GMAT Logic lesson, there are

important recurring themes that you will see in most GMAT problems:

ThINk LIkE ThE TEsTMAkER

• Abstraction

• Reverse Engineering

• Large or Awkward Numbers

• Exploiting Common Mistakes

• Selling the Wrong Answer and Hiding the Correct Answer

Focus on recurring themes, not just underlying content.

Each book in the Veritas Prep curriculum contains four distinct sections:

1 Skillbuilder We strongly suggest that you complete each Skillbuilder lesson before class at your own pace, and return to the Skillbuilder when you

recognize a content deficiency through practice tests and GMAT homework problem sets

The Skillbuilder section will:

• Cover content that is vital to your success on the GMAT, but is best

learned at your own pace outside the classroom

• Allow you to review and/or relearn the skills, facts, formulas, and content

of the GMAT Each student will have his own set of skills that are “rusty” or even brand-new, and will find other items that come back quickly

• Vary in length significantly for each book, based on the number of

underlying concepts (For instance, the Advanced Verbal lesson does

not have a Skillbuilder because you are already building on the concepts

introduced in three previous lessons.)

2 Lesson The lessons are designed to provide students with maximum value

added from an instructor by:

• Doing in-class problems together (Learning by Doing), and

• Analyzing those problems for the recurring takeaways

With each problem, there will be a detailed explanation that will help you understand how the problem is testing a particular concept or series of concepts, what makes the problem hard, and what underlying skills are required to solve it

When relevant, there will be particular boxes for Think Like the Testmaker, Skills Meet Strategy, and Skillbuilder when you should be focused on

particular aspects of how the question is made or how the underlying content is being tested

N oTE: When doing in-class and homework problems, you should do your

work below the problem, and you should not circle the answer on the

actual question (just note it on the bottom of the page) That way, if you want to redo problems, you can simply cover up your work and proceed

How This Book Is Structured

3 You Oughta Know The You Oughta Know sections will round out each lesson

and cover:

• Obscure topics that arise infrequently

• More advanced topics that are not common on the GMAT but do get

tested

While these uncommon content areas do not warrant in-class time, we

believe you should have some exposure to these topics before taking the

GMAT Therefore you should complete these sections before moving to

the homework problems As with the Skillbuilders, the length of these will

vary depending on their importance

4 Homework Problems In many ways, the homework problems are the most

important part of each book After refreshing core content in the Skillbuilder

and then applying that knowledge in the lesson, you must reinforce your

understanding with more problems

Each question is accompanied by a detailed explanation in your online

student account, as well as a quick-reference answer key on the last page

A majority of questions are above the 50th percentile in difficulty, and they

are arranged in approximate order of difficulty (easiest to most difficult) By

completing all of the homework problems, you will learn all of the different

iterations of how concepts and skills are tested on the GMAT

Homework problems are designed to be challenging, so do not despair if

you are answering questions incorrectly as you practice! Your goal should

be to learn from every mistake Students can miss a significant percentage of

questions in each book and still score extremely high on the GMAT, provided

that they learn from each problem Embrace the challenge of hard problems

and the notion that every mistake you make in practice is one that you will

know to avoid on the GMAT when every question counts

As you will see in the lesson to follow, GMAT geometry strategy can be best epitomized

by the phrase “leverage your assets.” Do you remember proofs from high school? To

solve a geometry problem, you must use small and seemingly unimportant pieces of

information—a length here, an angle there—to build a broader picture This ability to

put together multiple pieces of information and leverage all the assets in a problem

is the core skill that the GMAT tests with geometry Like a consultant restructuring a

company to attain a higher ROI, you must use cleverly hidden but fairly basic geometry

concepts to solve seemingly impossible problems But you can’t leverage assets that

you don’t have, so in this Skillbuilder section you will see (and have an opportunity

to practice) the major rules, formulas, and concepts that you need to solve any GMAT

geometry problem

Lines and Angles

When we refer to “lines,” we are describing straight lines that never alter direction.Two lines that intersect form four angles, with two pairs of angles being identical on opposite sides of the intersection The sum of the two angles on one side of a line always is 180°

Parallel lines on a two-dimensional plane will never intersect.

Two perpendicular lines are at 90° to each other.

This symbol indicates that the two lines are perpendicular to each other.

While most students are comfortable with the concepts relating to lines and angles, they frequently forget to apply these rules on complex problems Consider the following figure from a problem in a geometry book:

Note that angle CED is equal to angle AEF because of intersecting lines , and that angle CDE equals angle EAF because of parallel lines intersected by a straight line It is very easy to overlook these facts and instead focus on other rules relating to triangles and quadrilaterals

A

EF

Lines and Angles Drill

Use the figure below to answer the following questions:

1 2 3 4

5 6 7 8

1 If the measure of angle 2 is 77°, what is the measure of angle 5?

2 If the measure of angle 2 is 77°, what is the measure of angle 1?

3 What is the sum of the measures of angles 1, 2, 5, and 6?

4 If ℓl and ℓ2 are parallel and the measure of angle 8 is 104°, what is the

8 If ℓ1 and ℓ2 are parallel and the measure of angle 1 is 90°, what are the

measures of every other angle?

8 If ℓl and ℓ2 are parallel and the measure of angle 1 is 90°, what are the measures

of every other angle?

90°

The sum of the angles in a triangle is always 180°, regardless of shape.

If you know two angles of a triangle you can always find the third angle:

The height or altitude of a triangle is defined as the distance from the base to the

opposing apex The altitude is always perpendicular to the base

Any side can be the base, and the area is the same for all base/height combinations in

the same triangle The longest side of a triangle is always the side opposite the greatest

angle The shortest side of a triangle is always the side opposite the smallest angle

Important Triangle Concepts:

1 Right Triangles: A right angle triangle (one angle equals 90°) allows us to

calculate the length of one side when we know the length of two other sides,

by using a2 + b2 = c2.Right triangles are an integral part of geometry on the GMAT and must be mastered by students In this section we will look at some of the common right triangles and how they are tested

2 Right Triangles with Certain Sides: While students must be prepared to deal

with any combination of sides with right triangles, there are a couple specific right triangles with whole number sides that are frequently used on the GMAT The right triangles below, with sides 3, 4, 5, and 5, 12, 13, are those commonly used triangles Note that the sides of these triangles may be any multiple of the side given in these example (e.g 6, 8, 10 or 2.5, 6, 6.5, etc.)

a = 3

4

Because one angle is 90° we can use the Pythagorean Theorem: a2 + b2 = c2

Substitute the variables with their known values:

3 Right Triangles with Certain Angles: Two other very common right triangles are

ones with the angles 30, 60, 90 and 45, 45, 90 These triangles are unique because

you can determine all of the sides when only one side is given (traditional

pythagorean theorem requires that two sides are given)

4 30-60-90 Triangles: A triangle with the angles 30, 60, and 90 always has sides

with lengths of the proportions 1, √ 3, and 2 Note that 2x is the longest side,

so 2x belongs on the hypotenuse Similarly, x is the shortest side and belongs

on the side opposite the 30° angle x√ 3 , as the length of the medium side,

belongs opposite of the 60° angle

Example: In the following two triangles give the value of side y.

In the first triangle we must divide the side opposing 60° by √ 3 to determine

the short side (the side opposing 30°) After this division and removing the

root from the denominator we see that y = 9 √ 3

In the second triangle, we must use the same procedure to determine the

5 45-45-90 Triangles (aka Right Isosceles Triangles): A triangle with the angles

45, 45 and 90, always has sides with lengths of the proportions 1, 1, and √ 2 :

Example: In the following triangle, what is the value of side y?

To determine the short sides of a 45-45-90 triangle, we must divide the hypotenuse by √ 2

√

_

32

√ 2 = √ _ 32 _ 2 = √ _16 = 4 Therefore y = 4 in this example

N oTE: A 45-45-90 triangle is simply one half of a square Therefore when determining the diagonal of a square, use the relationship you have learned with the 45-45-90 triangle

6 Isosceles Triangles: An isosceles triangle has at least two sides and two angles

that are the same

When an altitude is drawn in an isosceles triangle from the unequal side, it

always creates two equal triangles In other words, the altitude splits the base

and the angle evenly when drawn from the unequal side In the following

diagram, isosceles triangle ABC is split by an altitude drawn from the vertex

at point B to point D on unequal side AC Because of this rule we know that

AD = DC, angle ABD = angle DBC and triangle ABD is congruent (equal) to

triangle DBC

A

B

CD

7 Equilateral Triangles: In an equilateral triangle, all sides are of equal length and

all angles are 60°

Since an isosceles triangle is one with at least two sides and two angles equal, every equilateral triangle is also an isosceles triangle Equilateral triangles are very common on the GMAT and possess several important properties that students need to memorize

When you draw in the height of an equilateral triangle, two 30-60-90 triangles are formed Using the 30-60-90 ratios we just learned, you can deduce that the height of an equilateral triangle will always be √

3 _ 2 times the side of the triangle

As with all triangles, the area of an equilateral triangle is 1 2 Base x Height

60°

Quadrilateral literally means “four-sided” The angles in a quadrilateral always add up

to 360° The four most frequently encountered types of quadrilaterals on the GMAT are:

3 Parallelogram

a Opposite sides are parallel and of the same length

b The interior angles add up to 360°

h = height

Area: b ∙ h Perimeter: 2a + 2b

4 Trapezoid

a Two sides are parallel

b The interior angles add up to 360°

Area: 1 2 (b + c) ∙ h Perimeter: a + b + c + d

Because you can’t use either b or c as the base, we find the average of the two when

calculating the area of a trapezoid

1 What is the area of a square with perimeter 20?

2 What is the width of a rectangle with area 15 and length 5?

3 If a parallelogram has area 30, perimeter 26, and two of the sides have length 6,

what is the length of each of the other two sides?

4 If a trapezoid has area 24, height 6, and one of the parallel sides has length 3, what

is the length of the other parallel side?

5 If a quadrilateral has four 90° angles and sides of length 3, 3, 8, and 8, it is which of

the following figures: square, rectangle, parallelogram, trapezoid?

6 If three angles of a quadrilateral have measures 35°, 145°, and 35°, is it a parallelogram?

7 If all of a quadrilateral’s sides have length 2, is it a square?

8 If all of a quadrilateral’s sides have length 2 and its area is 4, is it a square?

3 If a parallelogram has area 30, perimeter 26, and two of the sides have length

6, what is the length of each of the other two sides?

7

4 If a trapezoid has area 24, height 6, and one of the parallel sides has length 3, what is the length of the other parallel side?

5

5 If a quadrilateral has four 90° angles and sides of length 3, 3, 8, and 8, it is which

of the following figures: square, rectangle, parallelogram, trapezoid?

It is a rectangle, parallelogram, and trapezoid, but it is not a square

6 If three angles of a quadrilateral have measures 35°, 145°, and 35°, is it a parallelogram?

Not necessarily (The two 35° angles aren't necessarily opposite each other; the figure might be a trapezoid.)

7 If all of a quadrilateral’s sides have length 2, is it a square?

Not necessarily (it could be a rhombus)

8 If all of a quadrilateral’s sides have length 2 and its area is 4, is it a square?

Yes

As in arithmetic, it is essential that students are confident with their definitions in

geometry Let’s discuss important circle definitions:

1 Radius: The radius of a circle describes the distance from the center of a circle to

the circle itself

2 Diameter: The diameter of a circle describes the distance from one side of the

circle to the other side, intersecting the center of the circle The diameter is twice

the length of the radius

3 Chord: A line that connects any two points on a circle is known as a chord The diameter of a circle is an example of a chord.

4 Circle Formulas: π (pronounced pie, spelled “pi”) is a symbol that describes

a number that is essential when solving mathematical problems involving circles π is the ratio of the circumference of a circle to the diameter of that circle The circumference is the distance around a circle (eqivalent to the perimeter of a polygon) π = 3.14.* Since fractions often simplify arithmetic on the GMAT, knowing that π can also be expressed as 22 _ 7 may be helpful.

1 Area: πr2 (where r = radius)

2 Circumference: 2πr or πd (where d = diameter)Example: If the circumference of a circle is x, then express the area of that circle

in terms of x

If 2πr = x then we can express r in terms of x r = x _ 2π With one more substitution

we can see that A = π ( x _ 2π )2 Simplifying further, we see that A = πx _ 4π22 By canceling the π from the top and the bottom the final answer is A = x _ 4π 2

* π is an irrational number, as it has an infinite number of decimal places However, for the purposes of the GMAT, it is sufficient to know the first two decimal places

5 Central and Inscribed Angles: A central angle is any angle whose vertex

(point of origin) is at the center of the circle An inscribed angle is any angle

whose vertex (point of origin) is on the circumference of a circle

and

6 Sector: A sector is a portion of a circle defined by two radii and an arc carved

by a central angle In the following diagram, the shaded section would be

described as sector ABCD

C

B D

A

(center) Sector ABCD

7 Tangent: A line that touches a circle at only one point on a circle is called a

tangent The tangent is perpendicular to the radius at the point of tangency

tangent

20°

Additional formulas and Properties of Circles

1 Arcs and Central Angles: Some of the more important properties of circles

deal with the relationship between central angles and their corresponding arcs Consider the following diagram:

In this diagram we see that the central angle cutting out minor arc AB is equal

to 40º There is a direct relationship between that central angle and the arc that the angle subtends (cuts out) Because the central angles of a circle total 360º and the central angle in this example is 40°, we know that the minor arc

AB must be 40 360 , or 1 9 th of the total circumference

In any circle, the arc that is subtended (cut out) by a central angle relates to the circumference in the same proportion that the central angle relates to 360º

Central Angle

360 =

Minor Arc

Circumference

40°

A B

In addition to the unique relationship between arcs and their central angles, there is

also an important relationship between central angles, inscribed angles, and arcs

2 Inscribed Angles:All inscribed angles that subtend (cut out) the same arc or

arcs of equal length are equal in measure In the following diagram, angles x, y,

and z are all equal because they are inscribed angles that cut out minor arc AB:

x° = y° = z°

3 Inscribed Angles and Central Angles:Any inscribed angle that cuts out the

same arc as a central angle is exactly one-half the measure of that central

angle In the following diagram, angles x, y, and z must all be 20° because they

are cutting out the same minor arc AB as the 40° central angle

4 Inscribed Angles and Arcs:Because there is a distinct relationship between inscribed angles and central angles, we can use our previous knowledge to determine any arc that is cut out by an inscribed angle

To determine an arc from an inscribed angle, simply draw in the central angle (which will always be twice the measure of the inscribed angle) and use the arc/central angle proportion to determine the length of the arc

In the following figure, what is the length of minor arc AB?

With an inscribed angle of 30 degrees, that arc takes up 1 6 of the circle’s circumference As the circumference is 12π, the arclength is 2π

30° r = 6

1 If a circle has a radius of 2, what is its area?

2 If a circle has a circumference of 24π, what is its area?

3 If a circle has an area of 81π, what is the length of the circle’s longest chord?

4 Circle A has a radius of 7 and circle B has a circumference of 15π Which is the larger

circle?

5 If a circular pizza pie with a diameter of 16 inches is cut into eighths, what is the

area of each of the slices?

6 If a circle has an area of 9π, which of the following could be the lengths of chords

on that circle: 1, 3, 4, 6, 7, 8?

7 If a car’s tires cover 20π inches for every revolution, what is the outer diameter of

each of the tires?

8 If 20 circular pepperoni slices each with a 1-inch diameter cover a circular pizza

pie with diameter 14 inches, what fraction of the pizza is covered with pepperoni

slices?

9 If a circle centered on point O has an area of 36π, what is the length of minor arc AC

if angle AOC is 60 degrees?

10 In the figure below, Y marks the center of the circle The length of chord AB is 9

and the length of chord BC is 12 What is the circumference of circle Y?

A

B

CY

7 If a car’s tires cover 20π inches for every revolution, what is the outer diameter

of each of the tires?

20 inches

8 If 20 circular pepperoni slices each with a 1-inch diameter cover a circular pizza pie with diameter 14 inches, what fraction of the pizza is covered with pepperoni slices?

5 _

49

9 If a circle centered on point O has an area of 36π, what is the length of minor arc AC

if angle AOC is 60 degrees?

2π

10 In the figure below, Y marks the center of the circle The length of chord AB is 9 and the length of chord BC is 12 What is the circumference of circle Y?

Coordinate geometry is an area of increasing importance on the GMAT Students are

reporting that they have faced multiple questions on coordinate geometry on their

tests Let’s go over the basic definitions and properties of the coordinate geometry

plane:

The following figure is called a coordinate plane

Definitions for the Coordinate Geometry Plane

1 The coordinate geometry plane has four distinct quadrants that are labeled

above and go from I to IV in a counterclockwise direction

2 The horizontal line is the x-axis

3 The vertical line is the y-axis

4 The intersection of the x-axis and y-axis is the origin

5 Every point on the coordinate plane can be described by an ordered pair (x,

y), where x describes where the point is on the x-axis and y describes where

the point is on the y-axis The signs of x and y determine in which quadrant

the point will lie

1-2

Lines in the Coordinate Geometry Plane

All algebraic equations that are linear (have no exponents greater than 1) can be mapped on the coordinate geometry plane as a straight line It is easiest to map that line on the coordinate geometry plane when the equation is in the following form:

y = mx + b

1 Slope

In the equation y = mx + b, m describes

the slope of the line

For positive lines, the greater m is, the

steeper the line This can be seen by

setting b = 0 and trying different values for

m If m = 1, 2, or 3, then we have equations

y = x, y = 2x, and y = 3x, respectively When

x = 0, y = 0 in all of these equations But

when x = 1, the equations with the bigger

slope (the higher values of m) will have

higher values for y, as seen in the diagram

at right The slope of a line can be found

If the slope has a positive value, the line will be pointing up to the right.

If the slope has a negative value, the line will be pointing down to the right.

What is the slope of the line in this diagram?

change in y coordinatechange in x coordinate = slope _

10 - (-20) 20 - (-40) = 30 _ 60 = 1 2 = 0.5

10 20 30-30 -20

-40

10203040

-20

(20, 10)

(-40, -20)

1 2 3-3 -2 -1

1234

-4-3

m = 0

m = 1

m = 2

m = 3

The higher b is, the higher up the line intersects the y-axis Again, this can be seen by

setting m = 1 and x = 0, and trying different values for b

The point where the line intersects the x-axis is called the x-intercept The x-intercept

can be found by setting y = 0

The equation y = mx + b descibes a straight line with the following properties:

y-intercept = b

x-intercept = - b m

slope = m

1234

-4-3

1 2 3-3 -2 -1 m = 0

b = 0

b = 2

b = 3

b = -4

Important Properties and formulas for Difficult Coordinate

Geometry Problems

Now that we have reviewed the basic definitions and equations for use on the

coordinate geometry plane, let’s look at some important properties and formulas that allow you to solve even the hardest coordinate geometry questions quickly

1 Distance formula

To find the distance between two points on the coordinate plane, you will need to use the Pythagorean Theorem you learned in the Triangles section

To find the distance from the origin (which is point 0,0) to A, we take advantage of the fact that the line between the two points forms the hypotenuse in a right angled triangle

We see that one side has a length of 3 and the other has a length of 2, and we know that 22 + 32

= c2.Therefore c = √ 22 + 32 = √ _13 and the distance between the two points is determined

The work done above can be simplified with a general formula for finding the distance between any two points: √—–—–—–—–—(x1– x2) 2 + (y1 – y2) 2

Example: What is the distance between points (3,8) and (9,16) on the coordinate geometry plane?

By using Pythagorean theory, you can see that the triangle formed with these points is

a 6, 8, 10 right triangle so the missing piece—the hypotenuse—is 10 Or you can plug into the formula above to get the same result

1-2

1

-3

A(3, 2)