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Regents Review #7
Need to know how to construct:
1) a perpendicular bisector of a given line
segment
2) a bisector of a given angle
3) a line perpendicular to a given line
through a given point on the line
4) Congruent Angles:
5) a line perpendicular to a given line
through a point not on the given line
6) a line parallel to a given line through a
given point not on the given line
7) an equilateral triangle
Locus:
Locus: The set of all points that satisfy a given condition or set of conditions.
(represented by dashes)
The locus of points equidistant from two points:
The perpendicular bisector of the segment determined by the two points.
The locus of points equidistant from two parallel lines:
A line parallel to the given lines and midway between them
The locus of points equidistant from two intersecting lines:
A pair of lines that bisect the angles formed by the intersecting lines
The locus of points that are d distance from a line:
A pair of lines, each parallel to the given line and at the fixed distance d from the
given line.
d
d
The locus of points that are d distance from a fixed point.
A circle whose center is the fixed point and whose radius is the fixed distance d.
d
Compound Loci – locus of points that satisfy two or more conditions
Steps:
1) Determine the locus of points that satisfy the first condition. Sketch a diagram to show
these points.
2) Determine the locus of points that satisfy the second condition. Sketch a diagram to
show these points.
3) Locate the points, if any exist, that are common to both loci.
Two ways to graph a line:
#1 Chart (or table) method:
1) Solve the equation for y in terms of x. (y = mx + b form)
2) Choose any five values of x.
 (choose a couple of negative and positive values).
3) For each value of x, find y.
4) Plot each set of coordinates on the graph.
5) Connect the points and LABEL!!!
#2 Graphing using the slope-intercept method (y = mx + b)
1) Step #1: Solve the equation for y
2) Step #2: Find the slope of the line (coefficient of x)
3) Step #3: Find the y-intercept of the line (constant)
 (The y-intercept is the y-coordinate of the point where the line
intersects the y-axis.)
4) Step #4: Plot the y-intercept of the line.
5) Step #5: Use the slope to find two or more points on the line.
 (move from the y-intercept using the slope).
6) Step #6: Connect the points and LABEL!!!
Quadratics:
The graph of a quadratic equation is called a parabola.
The standard form of a parabola is y = ax2 + bx + c
Steps to graphing a quadratic equation:
1) Put the equation into standard form: y = ax2 + bx + c
b
2) Find the axis of symmetry using the formula: x 
 2a
3) Plug the quadratic into y = on the calculator.
4) Press 2nd graph to see the table. Copy down the table including the axis of
symmetry and at least 3 points above and 3 points below.
5) plot and connect the points and label the parabola.
Circles:
The standard form of a circle is: (x – h)2 + (y – k)2 = r2
The center is: (h, k)
The radius is: r
Steps to graphing a circle:
1) Find the radius and center of the circle from the standard form
2) Point the center.
3) Count up, down, left, and right the radius. Mark these points, they are your guide
points.
4) Sketch the circle passing through these guide points or use a compass to draw a
circle through these points.
To find the equation of a circle given the two endpoints of a diameter:
1) Find the midpoint of the two given points to get the center of the circle.
2) Find the distance between the center point (the midpoint) and one of the endpoints
of the diameter to find your radius.
3) Plug the center and the radius into the standard form of a circle.
To find the equation of a circle given the center and a point on the circle:
1) Find the distance between the center and the point on the circle to find the radius.
2) Plug the center and the radius into the standard form of a circle.
Systems:
Quadratic-Linear: a line and a parabola on the same graph
Steps:
1) Graph the line and label.
2) Graph the parabola and label.
3) Find the points of intersection and state what they are.

These are your solutions

There can be 0, 1, or 2 solutions (points of intersection)
Circle-Line: a circle and a line on the same graph
Steps:
1) Graph the line and label.
2) Graph the circle and label.
3) Find the number of points of intersection.

These are your solutions
There can be 0, 1 (tangent), or 2 (secant) points of intersection.
Part I:
______1) The locus of points equidistant from the sides AB and AC in scalene ΔABC
shown below is
A
(1) a median from A to CB
(2) an altitude from A to CB
(3) an angle bisector of A
(4) none of the above
C
B
______2) Which of the following is an equation of the locus of points equidistant from the
graphs of the equations x = 6 and x = -2?
(1) y = 2
(2) x = 2
(3) y = 4
(4) x = 4
______3) What is the slope of the line that is the locus of points equidistant from the
points (1, 7) and (5, -5)?
(1) -3
(2) 
1
3
(3) 3
1
(4)
3
______4) Points A and B are 5 units apart. How many points are there that are equidistant
from both points and 7 units from B?
(1) 1
(2) 4
(3) 2
(4) 0
______5) What is the locus of points that are equidistant from the point (-1, 1)?
(1) a parallel line
(2) a perpendicular line
(3) a circle
(4) an angle bisector
______6) The locus of the center of a school bus wheel that is moving down a straight
level road is:
(1) one line
(2) one circle
(3) two lines
(4) two circles
______7) What is the locus of points equidistant from a line?
(1) a parallel line
(2) a circle
(3) a perpendicular line
(4) two parallel lines
______8) The diagram below shows the construction of a line parallel to a given line.
Which of the following must be true?
(1) ESL  SRN
(2) ELS  SNR
(3) SL = RN
(4) ES = SR
______9) A city wants to place a water tower exactly 10 miles from the center of the
city. They want it to be built alongside the New York Thruway, which runs 8 miles from the
center of the city. How many locations can the city planner choose from?
(1) 0
(2) 2
(3) 1
(4) 3
______10) How many points are 3 centimeters from a given line and 4 centimeters from a
point on the line?
(1) 0
(2) 2
(3) 3
(4) 4
______11) Wilson got lost hiking in the woods. A rescue team found his campsite and
estimated that he could have traveled up to 3 miles since morning. What is the locus of
points in which the rescuers should search?
(1) a line 3 miles long
(3) a square 3 miles long and wide
(2) a circle with a 3 mile radius (4) a ray 3 miles from the campsite
______12) Which point is included in the locus of points located 3 units from the x-axis
and 7 units from the y-axis?
(1) (0, 7)
(2) (0, 3)
(3) (7, 3)
(4) (3, 7)
______13) The distance between points P and Q is 6 units. How many points are
equidistant from P and Q and also 3 units from P?
(1) 1
(2) 2
(3) 3
(4) 0
______14) What is the total number of points that are equidistant from two parallel lines
and also equidistant from two points on one of the lines?
(1) 1
(2) 2
(3) 3
(4) 4
_____15) Which graph could be used to find the solution to the following system of
equations?
y = x2
y=2
_____16) The graphs y = -x2 + 2x + 4 and x + y = 4 are drawn on the same set of axes.
How many solutions are there to this system of equations?
(1) 0
(2) 1
(3) 2
(4) infinitely many
_____17) The points (0, -5) and (5, 0) are solutions to which of the following sets of
graphs?
(1) y = x2 + 5 and y = x
(2) y = x2 - 10x + 25 and x + y = 5
(3) (x – 5)2 + y2 = 25 and y = 5x
(4) x2 + y2 = 25 and x – y = 5
Part II:
18) Write an equation of the locus of points equidistant from the graphs of the equations
of x = 4 and x = -2.
19) Show graphically the number of points 3 units from the coordinate (-3, 2) and 1 unit
from the graph of y = 2. How many points satisfy both loci?
20) Show graphically the number of points that are equidistant from the x- and y-axis and
3 units from the origin.
21) Construct a parallel line to AB through P.
P
•
B
A
22) Construct an equilateral triangle ABC using AB as one side.
A
B
Part III:
23) Given: point P on line m
a) Describe fully the locus of points 4 inches from line m.
b) Describe fully the locus of points d distance from point P.
c) Find the number of points that simultaneously satisfy the conditions in parts (a)
and (b) above for the following value of d.
(i) d = 5
(ii) d = 4
(iii) d = 3
24)
a) Draw the graphs of x – 2y = 2 and 2y – x = 8 on the coordinate system provided below.
b) Determine the locus of points equidistant from the lines in part (a). Draw this locus on
the graph.
c) On the same set of axes, sketch
the graph of the circle
(x – 2)2 + (y – 2)2 = 16.
d) How many points of intersection do
the graphs in part (b) and part (c)
have in common?
25) Sketch the graph of the following equations and mark the intersection(s) with an X.
(x – 2)2 + (y + 3)2 = 25
y = -x + 5
Is the line a secant line or a tangent line?
26) Solve the system graphically.
2x2 + 2y2 = 50
x=5
Is the line a secant line or a tangent line?
27) Solve the system graphically.
x2 + y2 = 16
y=x–4
Is the line a secant line or a tangent line?
28) Solve the system graphically.
y + x = x2
x+y=4
29) Solve the system of equations graphically:
y + x2 = 2x + 4
3y + 6 = 3x