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Regents Review #3
Ratio and Proportion:
The ratio of two numbers, a and b, where b is not zero, is the number
a
.
b
a c

or a : b = c : d
b d
In a proportion, the product of the means equals the product of the extremes.
(Cross-multiplying) The extremes are a and d. The means are b and c.
A proportion is an equation that states two ratios are equal.
Mean Proportional: when the means of a proportion are equal.
x y
“y is the mean proportional between x and z”

y z
When given a word problem with a ratio, your let statements are the ratio multiplied by x.
Similar Triangles:
Similar Polygons: Two polygons are similar if a one-to-one correspondence between their
vertices such that:
1) All pairs of corresponding angles are congruent.
2) The ratio of the lengths of all pairs of corresponding sides are equal.
“Angles are congruent and sides are in proportion.”
Ratio of Similitude: the ratio of the lengths of corresponding sides of similar polygons
The ratio of the sides of similar triangles is equal to the ratio of the perimeters, medians,
altitudes and angle bisectors.
The ratio of the areas of similar polygons is the square of the ratio of the sides.
Midsegment Theorem: A line segment joining the midpoints of two sides of a triangle is
parallel to the third side and its length is one half the length of the third side.
Proving Similar Triangles:
If we wanted to prove that two triangles are similar we can show that:
AA~: Two angles of one triangle are congruent to two corresponding angles of the other.
SSS~: The ratios of all three sides of the triangles are equal.
SAS~: The ratios of two pairs of corresponding sides are equal and the corresponding
angles included between these sides are congruent.
If two triangles are similar then angles are congruent and sides are in proportion.
If the sides are in proportion, then the product of the means equals the product of the
extremes.
Right Triangle Proportions:
Theorem: If the altitude is drawn to the hypotenuse of a right triangle then the two
triangles formed are similar to the original triangle and to each other.
Theorem: If the altitude is drawn to the hypotenuse of a right triangle then the length of
each leg is the mean proportional between the whole hypotenuse and the projection
adjacent to the leg.
Theorem: If the altitude is drawn to the hypotenuse of a right triangle then the length of
the altitude is the mean proportional between the lengths of the projections.
C
Leg b
Leg a
Altitude
A Projection a D
Projection b
B
Hypotenuse
Special Right Triangles:
30°-60°-90° Special Right Triangle:
45°-45°-90° Special Right Triangle:
45°
30°
2
3
2
1
45°
60°
1
1
Coordinate Geometry:
Coordinate Geometry is the blending of algebra and geometry.
The Coordinate Plane is separated into four regions called quadrants.
Every point in the coordinate plane can be described by two numbers,
called the coordinates of Cartesian coordinates. They are written as an ordered pair.
The first number is called the x-coordinate or the abscissa.
The second number is called the y-coordinate or the ordinate.
If you have an equation of the form y = mx + b, m is the slope.
Slope Formula:
m
y 2  y1
x 2  x1
Parallel lines: same slope
positive
zero
no slope
Perpendicular lines: negative reciprocal slopes
negative
no slope
zero
Collinear points: Point that are on the same line.
If points A, B, and C are collinear then the slope of AB is equal to the slope of BC.
Writing the Equation of the Line:
Use y = mx + b when you have a slope and a y-intercept.
Use y – y1 = m(x – x1) when you have 2 points or a slope and one point.
Midpoint of a line segment is that point that divides the segment into two congruent
segments.
 x  x2 y1  y 2 
,

Midpoint Formula: M   1
2 
 2
The distance formula is the length of line segments or distance between two points.
Distance Formula:
d
x2  x1 2   y2  y1 2
When you need to write the equation to a perpendicular bisector to a given line segment,
find the midpoint of the line segment and this is your point. Find the slope of the line
segment and take the negative reciprocal to use as your slope.
Part I:
_____1) D and E are midpoints. Find AB.
(1) 8
(2) 32
(3) 16
(4) 64
_____2) The sides of a triangle are 3, 7, and 9. Find the length of the shortest side of a
similar triangle whose longest side is 36.
(1) 28
(2) 12
(3) 9
(4) 6
_____3) Find x in the accompanying diagram:
(1) 2.5
(2) 7.5
(3) 5 2
(4) 2 5
_____4) In ΔABC, angle C is a right angle, altitude CD is drawn to the hypotenuse AB and
CD = 12. The lengths of the segments of the hypotenuse cannot be:
(1) 3 and 48
(2) 7 and 18
(3) 10 and 14.4
(4) 12 each
_____5) In the diagram, DE is parallel to AC. BD = 4, DA = 6 and EC = 8. Find BC to the
nearest tenth.
(1) 13.3
(2) 4.3
(3) 5.3
(4) 8.3
______6) When the coordinates of A are (2, -3) and of B are (2, 7), the length of AB is
(1) 4
(2) -4
(3) 10
(4) -10
_____7) At a certain time of the day, the shadow of a 5' boy is 8' long. The shadow of a
tree at this same time is 28' long. How tall is the tree?
(1) 17.5’
(2) 8.5’
(3) 16’
(4) 20’
_____8) As marked, by which method would it be possible to prove these triangles similar
(if possible)?
(1) AA~
(2) SAS~
(3) SSS~
(4) not similar
_____9) Given AC = 42, CB = 46, AB = 48. D, E, and F are midpoints. Find the perimeter of
ΔDEF.
(1) 34
(2) 68
(3) 48
(4) 136
_____10) What is the slope of the line containing the points (-5, 2) and (7, 4)?
(1)
1
6
(2) 
(3) 
1
6
9
5
(4) -6
_____11) The coordinates of two points are (0, 6) and (3, 0). The equation of the line
through these points is
(1) y = 2x + 6
(2) y = 
1
x+3
2
(3) y =
1
x+3
2
(4) y = -2x + 6
_____12) Which equation represents the line perpendicular to the line y =
(1) y = -5x - 3
(2) y = 
2
x-6
5
(3) y = 
5
x + 2?
2
5
x-6
2
(4) y = 5x + 3
_____13) The coordinates of the midpoint of line segment CD are (2, 6). If the
coordinates of point C are (1, 5), what are the coordinates of point D?
3
(1) ( , 1)
2
(2) (3, 7)
3 11
(3) ( , )
2 2
(4) (0, 4)
Part II:
14) In ΔABC, DE is parallel to AC, DE = 3 and AC = 8. If BD = 6, find the measure of AB.
B
D
A
E
C
15) In right triangle ABC, CD is the altitude to the hypotenuse, AB. The segments of the
hypotenuse, AB, are in the ratio of 1:4. The altitude is 6. Find the two segments of the
hypotenuse.
16) Write the equation of a line that is parallel to the line y = 2x – 5 and has a y-intercept
of 3.
17) Write the equation of a line that is perpendicular to the line y = 
2
x + 1 and passes
3
through the point (0,-5).
18) The endpoints of the diameter of a circle are A(4, 9) and B(-2, 15), what are the
coordinates of the center of the circle?
19) In ΔABC, the coordinates of A are (-2, -5), and the coordinates of B are (6, 3). Find,
in simplest form, the length of AB.
20) What is the equation of the line that passes through the points (1, 5) and (-1. -2).
Part III:
21) In triangle ABC, D is the midpoint of AB and E is the midpoint of BC. If AB = 10,
BC = 18 and the perimeter of triangle ABC = 48, find the DE.
B
D
A
E
C
22) Write the equation of the perpendicular bisector to the line segment whose endpoints
are (2, 2) and (6, 6).
23) Triangle ABC has coordinates A(4, 6), B(8, -2), and C(6, 10). Find the perimeter of the
triangle. Express your answer in simplest radical form.
Part IV:
24)
Given: AD = 6, DC = 3,
BE = 4, EC = 2
Prove: ΔCDE ~ ΔCAB
25)
Given: ACE, DCB, AB || DE
DC DE

Prove:
BC
BA
STATEMENT
REASON
STATEMENT
REASON