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Regents Review #2
CONSTRUCTIONS:
Constructions:
A geometric construction is a drawing of a geometric figure done using only a pencil,
compass, and a straightedge, or their equivalents.
A straightedge is used to draw a line segment but is not used to measure distance or to
determine equal distances.
A compass is used to draw circles or arcs of circles to locate points at a fixed distance
from a given point.
Need to know how to construct:
1) a perpendicular bisector of a given line
segment
2) a bisector of a given angle
3) Congruent Angles:
4) a line perpendicular to a given line
through a given point on the line
5) a line perpendicular to a given line
through a point not on the given line
Constructions and Triangles:
An altitude of a triangle is a line segment drawn from any vertex of the triangle,
perpendicular to and ending in the line that contains the opposite
side.
(use the construction for a line perpendicular to a given line [the
side of the triangle] through a point not on the line [the opposite
vertex])
The perpendicular bisector is any line or subset of a line that is perpendicular to the line
segment at its midpoint.
(use the construction for a perpendicular bisector of a given line
segment [the side of the triangle])
The median of a triangle is a line segment that joins any vertex of the triangle to the
midpoint of the opposite side.
(use the construction for a perpendicular bisector of a given line
segment [the side of the triangle] to find the midpoint of the
side then connect the midpoint and the vertex opposite the side)
An angle bisector is a line segment that bisects any angle of the triangle and terminates in
the side opposite that angle.
(use the construction for a angle bisector on the angle you wish
to bisect)
Points of Concurrency:
Concurrent – when lines intersect in one point
Equidistant – at equal distance
Circumcenter – the point where the three perpendicular bisectors of the sides of a
triangle are concurrent. (this point is the same distance from each vertex)
Orthocenter – the point where all three altitudes of a triangle are concurrent.
Centroid – the point where all three medians of a triangle are concurrent. (this point
divides the medians up into a ratio of 2:1)
Incenter – the point where all three angle bisectors of a triangle are concurrent. (this
point is the same distance from each side of the triangle)
TRIANGLE INEQUALITIES:
Properties:
Two sides of a triangle must add to be greater than the third side
An exterior angle of a polygon is an angle that forms a linear pair with one of the interior
angles of the polygon.
The measure of an exterior angle is equal to the sum of the measures of the nonadjacent
interior angles.
The measure of an exterior angle of a triangle is greater than the measure of either
nonadjacent interior angle.
The largest side of a triangle is across from the largest angle and the smallest side is
across from the smallest angle.
A whole is greater than any of its parts.
Transitive Property of Inequalities: If a, b, and c are real numbers such that a > b and
b > c, then a > c.
Inequality Postulates:
Substitution Postulate: a quantity may be substituted for its equal in any statement of
inequality.
- If a, b, and c are real numbers such that a > b and b = c, then a > c.
Trichotomy Postulate: Given any two quantities, a and b, one and only one of the following
is true:
a > b or
a = b or a < b
Addition Postulate of Inequalities:
- If equal quantities are added to unequal quantities, then the sums are unequal in
the same order.
o If a > b and c = d, then a + c > b + d.
-
If unequal quantities are added to unequal quantities in the same order, then the
sums are unequal in the same order.
o If a > b and c > d, then a + c > b + d.
Subtraction Postulate of Inequalities: If equal quantities are subtracted from unequal
quantities, then the sums are unequal in the same order.
- If a > b and c = d, then a – c > b - d.
Multiplication Postulate of Inequalities:
- If unequal quantities are multiplied by positive equal quantities, then the
products are unequal in the same order.
o If a > b and c = d, then ac > bd.
-
If unequal quantities are multiplied by negative equal quantities, then the
products are unequal in the opposite order.
o If a > b and –c = -d, then –ac < -bd.
Division Postulate of Inequalities:
- If unequal quantities are divided by positive equal quantities, then the products
are unequal in the same order.
a b
o If a > b and c = d, then 
c d
-
If unequal quantities are divided by negative equal quantities, then the products
are unequal in the opposite order.
a
b

o If a > b and -c = -d, then
c d
PYTHAGOREAN THEOREM:
In a right triangle, the sum of the squares of the legs is equal to the square of the
hypotenuse.
Pythagorean Theorem:
a2 + b2 = c2
(where a and b are the legs and c is the hypotenuse)
c
a
b
Converse of the Pythagorean Theorem: In a triangle, if the sum of the squares of the
two shorter sides is equal to the square of the larger side, then the triangle is a right
triangle.
Simplifying Radicals:
1) Find the largest perfect square that goes into the number underneath the radical
and how many times this perfect square goes into that number.
2) Write two new radical signs. Put the perfect square you found in step 1 underneath
the first radical sign and the other factor underneath the second radical sign.
3) Take the square root of the perfect square and leave the other radical alone.
*Make sure you use the biggest perfect square and that no other perfect squares go into
the number underneath the radical of your answer.*
Solving Quadratic Equations:
1) Put the equation into standard form: ax2 + bx + c
2) Factor
3) Set each factor = 0
4) Solve each equation
5) Check
Part I:
_____1) Which one of the following represents the lengths of the sides of a right
triangle?
(1) {8,15,17}
(2) {5,5,10}
(3) {7,8,12}
(4) {7,9,11}
_____2) The diagram below represents the relative position of three towns on a map. If
town B is 40 miles northeast of town A and town C is 60 miles southeast of town B, which
inequality best describes the distance between town A and town C.
B
North
West
40
60
A
C
East
South
(1) AC > AB + BC
(2) AC < AB + BC
(3) AC = AB + BC
(4) AC > AB and BC > AC
_____3) Find JN in the diagram below.
(1) 16 cm
(2) 4 cm
(3) 12 cm
51
(4)
cm
3
_____4) In triangle DEF, mD = 37º and mF = 56º. Find the measure of an exterior
angle at E.
(1) 87°
(2) 143°
(3) 93°
(4) 180°
_____5) The angles of a triangle are in the ratio of 1:3:5. Find the measure of the largest
angle of the triangle.
(1) 20°
(2) 80 °
(3) 60 °
(4) 100°
_____6) In circle O, diameter AB is perpendicular to chord CD at E. If OE = 6 and the
radius = 10, what is CD?
A
(1) 32
(2) 16
(3) 10
(4) 8
D
O
E
C
B
_____7) What is the measure of angle DBC according to the diagram?
(1) 25°
(2) 155°
(3) 125°
(4) 145°
_____8) Which of the following could represent the lengths of the sides of a triangle?
(1) {1, 2, 3}
(2) {6, 8, 15}
(3) {3, 4, 7}
(4) {5, 7, 9}
_____9) In triangle ABC, mA = 30º and mB = 50º. Which is the longest side of the
triangle?
(1) AB
(2)BC
(3) AC
(4) cannot be determined
_____10) If the side of a square is 5 cm, what is the length of the diagonal?
(1) 10
(2) 5 2
(3) 50
(4) 52
_____11) The angle bisectors of a triangle meet at a point that is:
(1) equidistant from the three sides of the triangle
(2) equidistant from the three vertices of the triangle
(3) outside the triangle
(4) do not meet at a point
_____12) In which triangle would the circumcenter, incenter, centroid, and orthocenter
be the same point?
(1) an isosceles triangle
(2) an equilateral triangle
(3) an isosceles right triangle
(4) none of the above
_____13) The altitudes of a triangle are concurrent at a point called the
(1) centroid
(2) incenter
(3) circumcenter
(4) orthocenter
_____14) If the height of a cone is 24 inches and the radius is 7 inches; what is slant
height of the cone?
(1) 31 inches
(2) 60 inches
(3) 25 inches
(4) 35 inches
_____15) Which point of concurrency is shown in the diagram?
(1) centroid
(2) incenter
(3) circumcenter
(4) orthocenter
_____16) The diagram below shows the construction of a line perpendicular to another
line. Which is the reason that PQ is perpendicular to line n?
(1) P and Q are equidistant from the two points on line n.
(2) Perpendicular lines form right angles.
(3) Two congruent triangles are formed.
(4) Point equidistant from a given point form a circle.
Part II:
17) If a rhombus has a short diagonal of 10 cm and a long diagonal of 24 cm; what is the
length of the side of the rhombus.
18) In ΔDEF, mD = 70° and DE > DF. Which angle is the smallest angle of ΔDEF?
19) In ΔACE, ED is the median to AC, CF is the median to AE and AH is the median to CE.
The medians meet at Q. If the measure of AQ is 24, find the measure of AH.
20) Construct the perpendicular bisector of line segment AB.
A
B
21) Construct the angle bisector for A.
A
22) Construct a perpendicular line to AB through P.
A
P
B
23) Construct a perpendicular line to AB through P.
•
P
A
B
24) Construct the altitude of ΔABC from C to AB.
C
A
B
25) Construct the median of ΔABC from A to BC.
C
A
B
26) Construct the angle bisector of B of ΔABC.
C
A
B
27) Construct the perpendicular bisector of AC of ΔABC.
C
A
B
28) Construct a similar triangle A’B’C’ to the given triangle ABC using line segment A’B’.
C
A
B
A’
B’
Part IV:
29)
Given: ΔABC is isosceles with base AC
Prove: BD > BA
STATEMENT
REASON