Coordinate Geometry

Question 1

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?

Answer 1

II

Question 2

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.

Answer 2

K(2) + 3K(1) = 17
5K = 17
K = 17/5

Question 3

equation of a horizontal line

equation of a vertical line

Answer 3

y = k (horizontal)
x = k (vertical)

Question 4

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?

Answer 4

x = 5

Question 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.

Answer 5

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

Question 6

slope for two points (x1, y1) and (x2, y2)

Answer 6

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

Question 7

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

Answer 7

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)

Question 8

slope of m=1

Answer 8

rise = run
slope triangle: 45-45-90 triangle
lines with slopes of m = 1 or m = -1 make 45° with the axes

Question 9

slopes of perpendicular lines are…

slopes of parallel lines are…

Answer 9

slopes of perpendicular lines are opposite-signs reciprocals

slopes of parallel lines are equal

Question 10

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?

Answer 10

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

Question 11

slope-intercept equation

Answer 11

y = mx + b

m: slope of the line
b: y-intercept of the line

Question 12

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.

Answer 12

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)

Question 13

What is the slope of the line with the equation 3x + 5y = 8?

Answer 13

solve for y to put it into slope intercept form
5y = -3x + 8
y = -3/5 x + 8/5
m = -3/5 x

Question 14

Line J passe through the points (-3, -2) (1,1) a,d (7,Q)

Find the value of Q

Answer 14

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

Question 15

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

Answer 15

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

Question 16

A circle in the x-y plane has a center of (6, 3) and a radius of 5. Find the two x-intercepts

Answer 16

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)

Question 17

equation of a circle with radius r centered at (0, 0)

Answer 17

x² + y² = r²

Question 18

reflections over the line y = x

Answer 18

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

Question 19

reflections over the line y = -x

Answer 19

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

Question 20

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?

Answer 20

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)

Question 21

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?

Answer 21

(4, 9)

Question 22

quadratic equation and parabola properties

Answer 22

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

Question 23

A parabola has an x-intercept at (-4, 0). If the vertex is at (2, 5), find the other x-intercept

Answer 23

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)