Coordinate Geometry Flashcards
Point M is the midpoint of segment AB.
If A = (2, -3) and M is on the negative x-axis, in what quadrant is B?
II
The equation of line M is Kx +3Ky = 17, for some nb K.
If line M passes through the point (2, 1), then find the value of K.
K(2) + 3K(1) = 17
5K = 17
K = 17/5
equation of a horizontal line
equation of a vertical line
y = k (horizontal) x = k (vertical)
Line P is a horizontal line. Line Q is perpendicular to line P, and passes through the point (5, 7). What is the equation of line Q?
x = 5
A rectatngle is formed by the lines y = 1, y = 4, x = 2, and line D. When the diagonal is constructed, it makes an angle of 30° with the base. Find the equation of line D.
line D is vertical so x = K where K > 0 give the vertices of the triangle letter names A(2, 1) B(K, 4) C(K, 1) this is a 30-60-90 triangle ratio long leg/short leg : AC/3 = √3/1 AC = 3√3 = K - 2 K = 2 + 3√3 equation of line BC: x = 2 + 3√3
slope for two points (x1, y1) and (x2, y2)
run = x2 - x1
rise = y2 - y1
slope m = rise/run = y2 - y1 / x2 - x1
for a slope m, we can find a point by going 1 to the right and m up/down
If a line goes through (2, -1) and has a slope of m = 5/3 find all the points (a,b) on the line where a and b are integers whose absolute values are less than or equal to 10
moving right : add 3 to x and 5 to y (2, -1) (5, 4) (8, 9) (11, 14) moving left: subtract 3 from x and 5 from y (2, -1) (-1, -6) (-4, -11)
(-1, -6) (2, -1) (5, 4) (8, 9)
slope of m=1
rise = run
slope triangle: 45-45-90 triangle
lines with slopes of m = 1 or m = -1 make 45° with the axes
slopes of perpendicular lines are…
slopes of parallel lines are…
slopes of perpendicular lines are opposite-signs reciprocals
slopes of parallel lines are equal
Line Q passes through (s, 0) and (0, s), for some nb s where s is unequal to 0. What is the slope of line Q?
if s is positive the rise and run have equal length, and the slope is negative
m = -1
if s is negative, the rise and run are still equal, and the slope is still negative
m = -1
slope-intercept equation
y = mx + b
m: slope of the line
b: y-intercept of the line
Find all the points (a, b) on the line y = (-4/3)x + 2 such that a and b are both integers with absolute values less than or equal to 10.
start at the y-intercept (0, 2) with the slope, 3 to the right and down 4 (3, -2) (6, -6) (9, -10) and then 3 to the left and up by 4 (-3, 6) (-6, 10)
What is the slope of the line with the equation 3x + 5y = 8?
solve for y to put it into slope intercept form
5y = -3x + 8
y = -3/5 x + 8/5
m = -3/5 x
Line J passe through the points (-3, -2) (1,1) a,d (7,Q)
Find the value of Q
algebraic solution m = 3/4 y = 3/4 x + b 1 = 3/4 (1) + b b = 1 - 3/4 = 1/4 y = 3/4 x + 1/4 Q = 7*3/4 + 1/4 = 11/2 =5.5
graphical solution cf video writing equations of lines
Find the distance between (-5, -1) and (3, 3)
x-leg = 8 y-leg = 4 we could do the Pythagorean theorem with 4 & 8 but it's easier to first scale down by a scale factor of k = 4 legs of 1 and 2 c² = 1² + 2² = 5 c = √5 scale back up by k = 4 distance = 4√5
A circle in the x-y plane has a center of (6, 3) and a radius of 5. Find the two x-intercepts
two radii go from the center to the two x-intercepts
each radius has a length of 5, and it’s the hypotenuse of a slope triangle
(6, 3) is 3 units above the x-axis
those are two 3-4-5 triangles
horizontal legs each have a length of 4
(2, 0) and (10, 0)
equation of a circle with radius r centered at (0, 0)
x² + y² = r²
reflections over the line y = x
line y = x has a slope of 1 and y-intercept of 0
it makes an angle of 45° with the x- and y-axes
set of all points in the x-y plane for which the x- and y-coordinates are identical
the image has the x- and y-coordinates switched
reflections over the line y = -x
line has a slope of -1 and y-intercept of 0
makes an angle of 45° with x- and y-axes
the image has the x- and y-coordinates switched, and each is given the opposite sign
Points J (5, 2) and K (-2, -5) are two vertices of an isosceles triangle. If L is the third vertex and has a y-coordinate of 4, what is the x-coordinate of L?
J and K are reflections over y = -x
so any point of that line would be equidistant from both J and K
L = (-4, 4)
A parabola with a vertex (2, 1) has a y-intercept of 9. What is the x-coordinate of the other point on the parabola with a y-coordinate of 9?
(4, 9)
quadratic equation and parabola properties
y = ax² + bx + c
when a > 0, parabola opens upward
when a < 0, parabola opens downward
when |a| > 1, parabola is skinny
when |a| < 1, parabola is wide
a quadratic equation can have two, one, or no solution
graphically, the parabola intersects the x-axes twice, once or not at all
A parabola has an x-intercept at (-4, 0). If the vertex is at (2, 5), find the other x-intercept
line of symmetry is x = 2
(-4, 0) is six units to the left of that line
the corresponding point, six units to the right
(8, 0)
