Coordinate Geometry

Reflections 1- What are the three common points reflections that could likely be asked on the GMAT? 2- And what are the three main reflections shapes that could be asked on the GMAT?

1- Reflections over the x-axis, reflections over the y-axis, and reflections over the origin 2- Reflecting shapes: ----> a line segment: simply reflect the endpoints of the line segment, then connect them together to create an image line segment. ----> a polygon: say the polygon is a triange, we simply need to reflect the vertices of the triangle and then connect them together to create an image triangle and a line. ----> a line: since a line does not have endpoints, pick any two points on the line and reflect them over the x-axis, the y-axis, or the origin. Once we have the two new points, we draw a line through them, and this new line is the reflection.

a line is parallel and higher to another line when...

1- both lines have the same slope 2- b in the higher line is larger than the lower line.

The x-intercept

At the x-intercept, the value of y is zero, and the coordinate pair is always (x, 0). To calculate the x-intercept, we must first isolate the variable x in the slope intercept equation to get: ⇒y=mx+b⇒y−b=mx⇒x=(y−b)/m Because we know that at the x-intercept, the value of y = 0, we can reduce the formula to: ⇒x=(0−b)/m⇒x=−b/m

The Information Needed To Define A Line so important. think deeply about this.

In summary, you can define a line if you know a point on that line in addition to one of the following: 1) The slope of the line, or the slope of a line that is parallel or perpendicular to the line. 2) A second point on the line (y-intercept, x-intercept, or any other point).

The Slope Of The Line And Steepness Of The Line

It seems to make sense to say the line with a larger slope is steeper than the line with a smaller slope; however, this is not quite correct. For example, it's true that a line with slope of 2 is steeper than a line with slope of 1, but it's not true that it is steeper than a line with slope of -3. It's because though the slope is the ratio between the rise and the run, both the rise and run can be negative. Therefore, the correct statement is: If the runs of two lines are the same, the larger the absolute value of the rise the steeper the line

All Points On A Line Are Connected By The Line's Slope-Intercept Equation

Let's assume that the equation for line L is y=1/2x+2. Any point that lies on line L must satisfy line L's equation. To satisfy an equation means to keep the equation in equality.

Parallel Lines

Parallel lines have the same slope and different y-intercept, and as a result, the lines will never intersect.

Using A Right Triangle To Calculate The Length Of A Line Segment Utilizing Right Triangles on a Coordinate Plane

The base of the triangle is 3 units, and the height of the triangle is 2 units. Thus, we can use the Pythagorean Theorem to determine the length of the hypotenuse

Slope Of A Line

The slope of a line provides us with a measure of the "steepness" of that line and can be calculated with the formula: where: →y2=thesecondy-coordinate →y1=thefirsty-coordinate →x2=thesecondx-coordinate →x1=thefirstx-coordinate →m=slopeoftheline →rise=theline'svertical(upanddown)movement →run=theline'shorizontal(leftandright)movement

Perpendicular Lines

The slopes of two perpendicular lines are negative reciprocals; negative reciprocals multiply to -1.

Question 6 - If u, w, and z are all integers, does line AB, with the equation y = wx + u, intersect line CD, which has the equation y = zx + z? 1) Line AB passes through the origin of the xy-plane and only crosses Quadrants I and III. 2) z < w

b my mistake was thinking that from 1, AB must intersect with CD since CD x intercept is -1. 0=xz+z --> x=-1 but what we really need to figure out the intersection is the slope of each.

Question 6 - "midpoint in disguise" Line segments MN and KL intersect on the xy-plane. If point M is located at (1, 10), and point K is located at (2, 4), what is the slope of line segment KL? 1) Line segments MN and KL bisect each other. 2) Point N is located at (13, 4).

c from 1, We should observe that because the lines are being bisected, the coordinate at the point of intersection will be the midpoint of either line segment. With statements one and two, we know that at the point of intersection of the two lines we have the midpoint of line segments KL and MN, and we also have two points from line segment MN. This means that we can determine the midpoint of line segment KL, and from there we will have two points on that line segment.

Question 10 - "finding the slope is enough" If line A is formed by the equation 4x = 3y - 15, the line formed by which of the following equations will never intersect line A? 4x = 2y + 3 8x = 3y - 1 8x = 6y - 2 12x = 6y + 1 4x = y + 4

c i didnt need to rephrase each formula to be in slope intercept to then test the same x-coordinate. In order to solve this problem, we have to go through each of the given equations and find the equation that has the same slope and different y-intercept; lines with the same slope and different y-intercept will never intersect because they are parallel. Neither the equation that we are given, 4x = 3y - 15 nor the equations in the answer choices are in slope-intercept form, so it will be easier to place them in such form prior to determining their slopes.

Question 12 - "not all visuals are correct" Circle O is plotted on the coordinate plane. If point A, which is located at (-3, y), and point B, which is located at (x, 4) are on the circumference of the circle, what is the value of y - x? 1) The center of circle O is at the origin of the coordinate plane. 2) x + y = 1

c my main issue is thinking that x,y has to be 0s since q says the origin is the center of the circle. however, this is not true. we gotta find it using the distance formula. key note is that both lines distances equal to each other Because we have two different points on the circle, we can actually set the distance between one point and the center equal to the distance between the other point and the center. Remember that all points on the circle are equidistant from the center of the circle

What is The Distance Formula and what is its shortcut?

d = √[( x₂ - x₁)² + (y₂ - y₁)²] this is basically the Pythagorean theorem, since you can stretch two lines from each point to form a right angle. The shortcut here is: 1) If two points have the same x-coordinates, then the distance between the two points can be obtained by the absolute value of the difference between their y-coordinates, or, by subtracting the smaller y-coordinate from the larger y-coordinate. 2) If two points have the same y-coordinates, then the distance between the two points can be obtained by the absolute value of the difference between their x-coordinates, or, by subtracting the smaller x-coordinate from the larger x-coordinate.

Question 9 - "what bisects does not necessarily perpendicular" What is the area of quadrilateral ABCD, which is graphed in the XY-plane? 1) ABCD is a quadrilateral with base AD parallel to the x-axis and coordinates A(2, 2) and D(6, 2). 2) Diagonals AC and BD bisect each other.

e i made the same mistake twice.thinkig that stmt 2 is saying perpinduclar "2) Diagonals AC and BD bisect each other."

The Slope Of A Line Can Be Positive, Negative, Zero, Or Undefined

for the undefined and zero slopes, think about the position of the zeros in the D or N in the slope formula

Other Reflections You May See on the GMAT

get back to this one, it is so rare, that i didnt bother for now.

Is having a point and B enough to find the equation?

i came up with this matrix. it is great .

The Midpoint Formula

midpoint(xm,ym) = (x₁+x₂)/2, (y₁+y₂)/2 where: →xm=the x-coordinate of the midpoint →ym=the y-coordinate of the midpoint →x1=the x-coordinate of point 1 →x2=the x-coordinate of point 2 →y1=the y-coordinate of point 1 →y2=the y-coordinate of point 2

if the slope of a line is negative, what quadrants does the line pass in?

must pass in Q2 and 4. if x and y intercepts are positive, then it passes Q1 as well. and if they're negative, it passes Q3.

the different forms of a line euqation

the slope intercept form y = mx + b the standard form Ax+By=C always better to change the form form standard to slope intercept ⇒Ax+By=C⇒By=−Ax+C⇒y=−Ax/B + C/B Notice that this is following the pattern of y = mx + b, and thus, we see that the slope is equal to −A/B and the y-intercept is equal to C/B.

Working With The Slope-Intercept Equation

there may be times when that form is obscured. Before we can work with the equation, it must be in the correct form! Often this will take some rearrangement of the equation's terms. For example, if we were given the equation 3x + 5y = 8, it may be tempting to conclude that the slope of this line is 3 and the y-intercept is 8. This would not be correct because the equation is not in slope-intercept form where the variable y is isolated

The Slope-Intercept Equation

⇒y=mx+b where: →y=they-coordinate for a point on the line →x=the corresponding x-coordinate for the point on the line →m=the slope of the line →b=the y-intercept of the line