Gmat - Geometry - Coordinate Plan, Lines And Angles

Front 1

What is the shortest distance btw 2 points?

Back 1

Straight Line - Angle:180°

Front 2

Do Parallel lines intercept each other?

Back 2

never!

Front 3

What are the Intersecting Lines Properties?

Back 3

• The sum of interior angles is 360

• Interior angles that combine to form a line sum 180°

• Angles found opposite each other where two lines intersect are equal

P.S. These properties apply to more than two lines that intersect at a point

Front 4

How is Coordinate Plan expressed?

Back 4

(a,b):a = x-coordinate, and b = y-coordinate

Front 5

How many types of slope exist?

Back 5

Front 6

Advanced Formulas and Functions

How to maximize linear functions in an Optimization problem?

Back 6

Since linear functions either rise or fall continuously, the max/min points occur at boundaries: at smallest or largest possible x as given in the problem.

Front 7

How do you plot a positive slope rise?

Back 7

rises upward from left to right

Front 8

How do you plot a negative slope rise?

Back 8

falls downward from left to right

Front 9

How do you plot a zero slope?

Back 9

Draw a horizontal line

Front 10

How do you plot an undefined slope?

Back 10

Draw a vertical line

Front 11

What are the slopes of x-axis and y-axis?

Back 11

The y-axis has an undefined slope

The x-axis has a Zero slope

Front 12

How is defined a slope?

Back 12

Theslope is defined as “rise over run”: (Y1 – Y2)/ (X1– X2)

Front 13

Rise and Run are similar along the entire line?

Back 13

NO!

In a line, any two other points have different “rise” and different “run”, but the slope will always be the same.

Front 14

What is the intercept of a line?

Back 14

• An intercept is a point where the line intersects a coordinate axis
• X-Intercept= (x,0)
• Y-Intercept= (0,y)
• To find x-intercepts, plug in 0 for y. To find y-intercepts, plug in 0 for x

Front 15

What is the Slope-intercept Equation?

Back 15

y =mx + b

m = slope

b = y-intercept

• For Horizontal and vertical lines

Horizontal: y = some number

Vertical: x = some number

Front 16

Step by Step on how to apply the Slope-intercept Equation from 2 points to a line.

Back 16

(1) Find the slope with rise over run

(2) Plug in the Slope

(3) Solve for b by plugging the coordinates of one point

(4) Write the equation in the form y = mx + b

Front 17

Find the Slope-intercept Equation

Point 1 = (5,-2)

Point 2 = (3,4)

Back 17

1. Rise over run: (-2 - 4) / (5 – 3) = -3
2. y = -3x + b
3. 4 = -3.3 + b --- b = 13
4. y = -3x + 13
   

P.S •If GMAT gives you one point and one intercept, don’t forget: an intercept is also a point, so you are able to find the equation.

Front 18

How to measure a distance between two points

Back 18

1. Draw a right triangle connecting points
2. Find the lengths of the 2 legs of the triangle
3. Use Pythagorean theorem
   

Front 19

How to calculate the distance of (x1,y1) from ax+by+c=0

Back 19

Front 20

Find the distance between (1,3) and (7,-5)?

Back 20

1. Firstleg = 6 (7-1)
   
2. Second leg = 8 (3- (-5))
   
3. Pythagorean:x² = 6² + 8² --- x = 10
   

Front 21

How to recognize the Quadrant that a line passes

Back 21

If you are required to find out which quadrant a given line passes to and you have the equation, set x and y to zero to find the two intercepts and draw the line.

Front 22

Do Parallel and perpendicular lines have same slope?

Back 22

YES!

Perpendicular lines have negative reciprocal slopes!!!

Front 23

A line is a Bisectors of another line if

Back 23

• The perpendicular has the negative reciprocal slope of the line segment it bisects.
  
• it cuts at 90º and into 2 equal lnghts

Front 24

To find the equation of a perpendicular bisector:

Back 24

1. Find the slope of the line segment
2. Find the slope of the perpendicular bisector (reciprocal)
3. Find the midpoint of AB
4. Find b for the bisector, by plugging the midpoint of AB
   

Front 25

Find the perpendicular bisector of line with points (2,2) and (0,-2)

Back 25

1. Slope= 2 - (-2) / 2 – 0 = 2;
2. Slope of the bisector = -1/2
3. Midpoint= (1,0)
4. Plugging:0 = -1/2 . 1 + b ---- b = ½
5. Equation of the perpendicular bisector: y= -1/2x + 1/2
   

Front 26

What are the reasons for two lines do not intersect one another?

Back 26

1. They are parallel no pair of numbers (x,y)will satisfy both equations.
2. Or the equations are the same.Then, infinitely many points will solve both equations.

• To test whether a point lies on a line,just test it by plugging the numbers into the equation.
  

Front 27

How to calculate the intersection point of 2 lines

Back 27

If lines intersect, both equations at the intersect point are true. That is, the ordered pair (x,y) solves both equations

lines y = 4x + 10 and 2x + 3y = 26 intersect?

replace y

4x + 10

in the second equation

–x= 4, y = 6

Front 28

How do you find the Co-ods of pts dividing line segments P1P2 in ratios

r/s?

Back 28

Front 29

2 non-vertical lines are parallel if...

Back 29

their slopes are parallel i.e. m1=m2

Front 30

2 non-vertical lines are perpendicular if...

Back 30

their slopes have a product = -1 i.e. m1.m2 = -1

Front 31

Rules to find the reflection of a point in the y-axis

Back 31

(i) Change the sign of abscissa i.e., x-coordinate.

(ii) Retain the ordinate i.e., y-coordinate.

Front 32

What is the formula for triangle area from coordinates?

Back 32

Hint 32

Observ the difference between y's and x's

Front 33

What is the importance of Discriminat in quadratic function graph?

Back 33

It shows the number of solutions/roots

• If (b² - 4ac) > 0, there are two real solutions or x-intercepts.
  
• If (b² - 4ac) = 0, there is one real solution or x-intercept.
  
• If (b² - 4ac) < 0, there are no real solutions or x-intercepts.
  

Front 34

How many roots does x^2 – 6x + 9 have?

Back 34

Do not simply assume an answer. Paper and pen always, even to ordinary questions.

Ans. one (repeated) real solution, zero inside the square root

Front 35

How many roots does x^2 + 3x + 3 have?

Back 35

two complex solutions

a negative number inside the square root

Front 36

How many roots does x2 – 2x – 3 have?

Back 36

two real solutions

a positive number inside the square root

Front 37

A quadratics function graph (y = ax² + bx+ c) is always a ....

Back 37

PARABOLA

• Ifa>0, the parabola opens upward.
  
• If a<0, the parabola opens downward.
  
• If|a| is large, the curve is narrow. If not, the curve is wide.
  

Front 38

Advanced Formulas and Functions

How to solve Maximizing quadratics functions in an Optimization problem?

Back 38

Make the squared expression equal to 0. Whatever value of x makes the squared expression equal 0 is the value of x that maximizes/minimizes the function

Front 39

Proportional functions

Back 39

For direct proportionality, set up ratios for the “before case” and the “after case” and equal them (regra de 3). The proportionality is defined by y = kx, where k is the proportionality constant.

Front 40

Inversely proportional functions

Back 40

For inverse proportionality, set up products for the “before” and “after” cases and equal them.

The inverse prop. formula is y = k/x

Front 41

If the amount of current and resistance are inversely proportional, and the current was 4 amperes and resistance is cut to one-third the original, what will be current?

Back 41

x1y1 = x2y2

3.4 = 1.x

x = 12