geometry Flashcards
positive slope will definitely pass through which quadrants.
- If the slope of line is negative, line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positives, line intersects the quadrant I too, if negative quadrant III.
- If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefor if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.
- Every line (but the one crosses origin or parallel to X or Y axis) crosses three quadrants. Only the line which crosses origin (0,0) OR is parallel of either of axis crosses two quadrants.
- The line with slope 0 is parallel to X-axis and crosses quadrant I and II, if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative.
In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis?
(1) a + b = -1
(2) The graph intersects the y-axis at (0, -6)
X-intercepts of the function f(x)f(x) or in our case the function (graph) y=(x+a)(x+b)y=(x+a)(x+b) is the value(s) of xx for y=0y=0. So basically the question asks to find the roots of quadratic equation (x+a)(x+b)=0(x+a)(x+b)=0.
(x+a)(x+b)=0(x+a)(x+b)=0 –> x2+bx+ax+ab=0x2+bx+ax+ab=0 –> x2+(a+b)x+ab=0x2+(a+b)x+ab=0.
Statement (1) gives the value of a+ba+b, but we don’t know the value of abab to solve the equation.
Statement (2) tells us the point of y-intercept, or the value of yy when x=0x=0 –> y=(x+a)(x+b)=(0+a)(0+b)=ab=−6y=(x+a)(x+b)=(0+a)(0+b)=ab=−6. We know the value of abab but we don’t know the value of a+ba+b to solve the equation.
Together we know the values of both a+ba+b and abab, hence we can solve the quadratic equation, which will be the x-intercepts of the given graph.
Answer: C.
slope concept
- The equation of the Y axis
- The equation of the X axis
- Slopes relation btw 2 lines
the slope of this line, 2, is greater than 4/3, the slope of the line drawn above. Hence, this line will be steeper than the line drawn above.
The slope of the line is 1. The slope of this line, 1, is less than 4/3, the slope of the line in the question. Hence, this line will be less steep than the line in the question stem
- is x = 0
- is y = 0
- A single point of intersection between two lines: a/m ≠ b/n 2.Distinct parallel lines: a/m = b/n ≠ c/p
- The same line: a/m = b/n = c/p
- Perpendicular lines: am = -bn
How to define graph of line
Any given line on the xy plane can be uniquely described using two characteristics – the line’s slope and a point through which the line passes.
Any given line on the xy plane can be uniquely described using two characteristics – the line’s slope and a point through which the line passes.
Question: Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?
Statements:
1.Slope of the line represented by ax + by – c = 0 is 2.
2.The line represented by ax + by – c = 0 passes through (-3, 0).
Ans = A
Question: If x and y are positive, is 4x > 3y? Statements:
1.x > y – x 2.x/y < 1
ANS = E
If ab≠0 and points (-a, b) and (-b, a) are in the same quadrant of the xy-plane, is point (-x, y) in the same quadrant?
(1) xy > 0
(2) ax > 0
Ans = c
First, a quick review of quadrants: what defines the quadrants are the +/- signs of x and y
1) In Quadrant I, x > 0 and y > 0
2) In Quadrant II, x < 0 and y > 0
3) In Quadrant III, x < 0 and y < 0
4) In Quadrant VI, x > 0 and y < 0
If (-a, b) and (-b, a) are in the same quadrant, that means that the x-coordinates have the same sign, and also the y-coordinates have the same sign. Look at the y-coordinates — if the two points are in the same quadrant, a & b have the same sign. They either could both be positive (in which case, the points would be in Quadrant II) or they could both be negative (in which case, the points would be in Quadrant IV).
Now, the question is: (-x, y) in the same quadrant as these two points?
(1) Statement 1: xy > 0
This tells us that x and y have the same sign — both positive or both negative. Now, we know a & b have the same sign, and x & y have the same sign, but there’s two possibilities for each, so we don’t know whether a & b & x & y all have the same sign. This is insufficient.
(2) Statement 2: ax > 0
This, by itself, tells us that a and x have the same sign – with this alone, we know that a & b & x all have the same sign, but we have zeor information about y. This too is insufficient.
Combined (1) & (2)
Prompt tells us a & b have the same sign. Statement #1 tells us x & y have the same sign. Statement #2 tells us x & a have the same sign. Put it all together –> we now know that x & y & a & b all have the same sign. Therefore, (-x, y) will have the same sign x- & y-coordinates as (-a, b) & (-b, a), and therefore all will be in the same quadrant. Combined statements are sufficient.
In the xy-plane, both line K and L intersect with axis-y. Is K’s intercept with axis-y greater than that of line L?
(1) K’s intercept with axis-x is greater than that of L.
(2) K and L have the same slope.
Why is C not the answer choice
https://gmatclub.com/forum/in-the-xy-plane-both-line-k-and-l-intersect-with-axis-y-is-94508.html
In the picture, quadrilateral ABCD is a parallelogram and quadrilateral DEFG is a rectangle. What is the area of parallelogram ABCD (figure not drawn to scale)?
(1) The area of rectangle DEFG is 8√5.
(2) Line AH, the altitude of parallelogram ABCD, is 5.
https://gmatclub.com/forum/in-the-picture-quadrilateral-abcd-is-a-parallelogram-and-127219.html
All inscribed angles that subtend the same arc are equal. The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle. Hence, all inscribed angles that subtend the same arc are equal.
