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Advanced Geometry Final Review & Answer Key
ANSWER KEY
1. Properties of a Rhombus
a. All sides are congruent
b. Diagonals are perpendicular
c. Diagonals bisect angles
d. It is a parallelogram
1
e. 𝐴 = 2 𝑑1 𝑑2
2. Properties of a Rectangle
a. Opposite sides are congruent and parallel
b. Angles are all 90 degrees
i. All congruent
ii. Consecutive angles are supplementary
c. Diagonals bisect each other
d. Diagonals are congruent and form 4 isosceles triangles
3.
a. Which have congruent diagonals?
i. Square and rectangle
b. Which have perpendicular diagonals?
i. Square and Rhombus
4. How do you find the measure of Interior Angles of a Polygon
a. 180(𝑛 − 2)
i. n = number of sides
5. What do the Exterior Angles total to?
a. 360 Degrees
6. How do you find the Median of a Trapezoid?
𝑏1 +𝑏2
a. ½ (top + bottom) OR
2
7.
What is a Secant?
Line slashes through
and touches twice
8. If a circle’s 𝐶 = 20𝜋 & 𝐴 = 40𝜋 𝑖𝑠 𝑖𝑡 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒?
a. False
What is a tangent?
Line touches the circle
Just once
9. What is the relationship between a chord and diameter?
a. A diameter is a chord. A chord isn’t always a diameter
10. An INCRIBED ANGLE (Pinched back) measure if 90 degrees, it intercepts an arc of 45 degrees. Is
this True or False
a. False
11. Write an equation in Point-Slop Form for the perpendicular bisector of the segment with the
endpoints L (4,0) & M(-2,3)?
a. 1st find the midpoint of the line
4+(−2) 0+3
i. ( 2 , 2 ) = (1 , 1.5)
b. 2nd Find the slope of the given line
3−0
3
1
i. 𝑚 = −2−4 = −6 = − 2
c. 3rd Then find the perpendicular slope
i. 𝑚 = 2 Opposite sign, reciprocal
d. 4th write the equation of the line by putting it into point-slope form
i. Answer: 𝑦 − 1.5 = 2(𝑥 − 1)
12. If the height of the triangle is 4cm and the area is 24cm2, what is the base?
1
a. ⊿𝐴 = (𝑏)(ℎ)
2
b.
1
24 = 2 (𝑏)(4)
i. 24 = 2𝑏
1. 𝑏 = 12𝑐𝑚
13. Find the sum of a regular convex dodecagon.
a. 180(𝑛 − 2) = 180(12 − 2) = 180(10) = 1,800
14. Find the measure of one interior angle of a regular polygon with 5 sides.
180(5−2)
a.
= 108
5
15. In an isosceles right triangle (2 sides equal) , if the hypotenuse is 12cm, what is the base?
12 = 𝑥√2
12
𝑥=
𝑥=
12
√2
12√2
2
𝑥 = 6√2
16. What is the area of an equilateral triangle that has a side length of 7cm?
a.
1
7 √3
2
2
𝐴 = (7) (
) =21.22
17.
5√3
If the measure of an altitude of an equilateral triangle is �√�
m. What is the perimeter?
𝑃 = 10 + 10 + 10 = 30
10
18. Given that line p is the perpendicular bisector of XZ; XY  4n, and YZ  14, find n.
a. 4𝑛 = 14 𝑛 = 3.5
b.
19. Suppose the chord of a circle is 24 inches long and is 16 inches from the center, find the length of
the radius.
a. 162 + 122 = 𝑥 2
i. 𝑥 = 20 inches is the radius
20. How many sides does a regular polygon have if the measure of one interior angle is 160 degrees?
180(𝑛−2)
a.
= 160
𝑛
b. 180𝑛 − 360 = 160𝑛
c. 20𝑛 = 360
i. 𝑛 = 18 sides
21. 𝑊ℎ𝑎𝑡 𝑎𝑟𝑐 (𝑠) 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 54 𝑑𝑒𝑔𝑟𝑒𝑒𝑠?
̂
a. 𝐵𝐶
̂
b. 𝐸𝐹
22. Find the measure of arc DE.
a. 63
23. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑟𝑐 𝐵𝐶𝐷.
a. 360 − 90 − 100 = 170
24. Find the Area and Perimeter of the figure?
a. 𝑃 = 6 + 4 + 2 + 3 + 4 + 7 = 26𝑐𝑚
b. 𝐴 = 36
25. Find the measure of one interior and one exterior angle of a hexagon.
a. (Regular) 1 interior- 120 1 exterior- 60
In triangle ABC, m⦟A = 90o, m⦟B=45o. If AB = 24 inches, find AC.
C
26.
AC=24
B
A
24
27. Find
a.
the area and perimeter.
𝐴 = 44 𝑃 = 34
28. If ABCD is a parallelogram, which sides are congruent?
a. AB and CD
BC and AD
29. 𝑥 = 8
𝑦 =6
Rectangle= 8(6)= 48
Trapezoid= ½ (8+16)(6)= 72
Whole figure = 48+72= 120
30. SV,TV, and UV are perpendicular bisectors of the sides ofPQR. Find RV.
a. 𝑅𝑉 = 26
31. Find HI, GE and mHGD.
a. 180 − 58 = 122 <HGD
b. 𝐻𝐼 = 9.1
c. GE=9.1
32. What is the exact measure of the diagonal of a rectangle with sides of length 2 inches and 3 inches?
a. 22 + 32 = 𝑥 2
i. 𝑥 = √13
33. The measure of an altitude of an equilateral triangle is 7√𝟑 in. Find the exact area.
1
a. 2 (7√3)(14) = 49√3
34. Find the exact area and perimeter of the figure.
1
a. Area= 192+192+96+2100𝜋 = 637. 08
b. Perimeter = 16+12+16+12+16+12+1/2(20𝜋)= 115.42
35. A chord of a circle is 20 inches long and 50 inches from the center of the circle. What is the radius
of the circle?
a. 102 + 502 = 𝑥 2
𝑥 = 10√26
36. Find the area of the figure.
1
a. 2 (10 + 18)(9)= 126
37-39- C is the center of the circle. AB is the diameter. ED is a tangent. AD and FD are secants.
37. Find DE/ED.
a. 122 + 𝐸𝐷2 = 302
i. 𝐸𝐷 = 6√21
38. Find mADF.
1
a. 2 (50 − 30) = 10
39. Find GF.
a. 42(18) = (20 + 𝑥)(20) = 17.8
40. How many sides does a regular polygon have if the measure of one interior angle is 144 degrees.
a. 10 sides
41. The aspect ratio of a TV screen is the ratio of the width to the height of the image. A regular TV
has an aspect ratio of 4:3. Find the height and width of a 60-inch TV screen to the nearest tenth of
an inch.
good question to look at for final discussed in class:
Tv is measured by length of diagonal= 60
The ratio is 4:3 width to height
Assign x to the width=4x and the height=3x
Use Pythagorean theorem to solve for x, and then plug back in to get the height and width.
(4𝑥)2 + (3𝑥)2 = 602
16𝑥 2 + 9𝑥 2 = 3600
25𝑥 2 = 3600
𝑥 2 = 144
𝑥 = 12
4(12)= 48 inches for width
3(12)=36 inches for height
42. Find the missing side lengths. Give your answer in simplest radical form. Tell whether the side
lengths form a Pythagorean Triple.
a. 32 + 92 = 𝑥 2
i. 3√10 = 𝑥
43. Find the measure of angle ABC
a. 180 − 34 − 82 = 64
44. What is the area of a triangle whose base is 12 inches and height is 14 inches?
1
a. 𝐴 = 2 (12)(14) = 84 𝑖𝑛2
45. Tell whether the measures can be the side lengths of a triangle. If so, classify the triangle as
acute, obtuse, or right. 15, 18, 20
a. 15+18>20
i. Two smaller sides have to be bigger than the biggest sides.. so yes it is a triangle
1. 152 + 182 ? 202
a. 549 > 400
b. ACUTE
46. Find the value of X.
a. 7 = 𝑥√2
i. 𝑥 =
7√2
2
47. Solve for X
a. 52 + 𝑥 2 = 182
i. 𝑥 = √299
48. Solve for X
1
a. 𝐴 = 2 (10 + 15)(8)
i. A= 100
49. Solve for X
a. 54 = 13𝑥
54
𝑥 = 13
50. Solve for X
a. 52 + 𝑥 2 = 102
i. 𝑥 = 5√3
1
1. 𝐴 = 2 (10)(10√3)
a. 𝐴 = 50√3
51. Find the area of triangle EFG
Area of triangle= ½ bh
Need height- Pythagorean thm
52 + 𝑥 2 = 132
25 + 𝑥 2 = 169
𝑥 2 = 144
x=12 which is the height of triangle
1
𝐴 = (10)(12)
2
𝐴 = 60 𝑢𝑛𝑖𝑡𝑠 2
52. What is a major Arc? Name one.
a. Major Arc- any Arc over 180 Degrees
̂ = 281
b. 𝐴𝐷𝐶
53. An ______________________ is a transformation that does not change the shape or size of a figure.
a. Isometry
54. Which types of transformations do not change the size or shape of the original figure?
a. Rotation, reflection, transformation
55. Find the area of a regular hexagon with a side length of 10 inches.
1
𝐴 = 𝑎𝑃
2
1
𝐴 = (5√3)(10 ∙ 6)
2
𝐴 = 80√3
30
𝑎 = 5√3
10
5
tan 36 =
Find the area of a regular pentagon with an apothem of 8 inches.
𝑥
8
8 tan 36 = 𝑥
56.
36
so 2 (8 tan 36)
8
1
𝐴 = (8)(5(16 tan 36)
2
𝐴 = 232.5
What is the area of a square in which each diagonal is 6m?
𝑥 2 + 𝑥 2 = 62
57.
2𝑥 2 = 36
6
2
𝑥 = 18
𝑥 = 3√2
58.
Find the measure of an inscribed angle if the central angle is 40.
a. 20
59. Find X
a. 𝑥 2 = (5)(5 + 7)
i. 𝑥 = 2√15
60. Find A
a. 𝑎(12) = 8(10)
i. 12𝑎 = 80
80
1. 𝑎 = 12
a. 𝑎 = 6.67
𝐴 = (3√2)(3√2) = 18𝑚2
61. Find the measures of a parallelogram if <D=140 & <4=18
a. < 1 = 22
b. < 2 =18
c. < 3 =22
d. < 4 =18
e. < 5 =140
62. Find the value of x in a parallelogram. <1=<3=3.5x, <2=3x-80
a. 3.5𝑥 − 3𝑥 − 80 = 180
i. 6.5𝑥 = 260
1. 𝑥 = 40
a. < 1 =140
b. < 2 = 40
c. < 3 =140
d. < 4 = 40
63. Find the values of x and y for parallelogram ABCD.
a. 3𝑥 − 40 + 𝑥 − 𝑦 = 180 and 2𝑦 = 𝑥 − 𝑦
4𝑥 − 𝑦 = 220
𝑎𝑛𝑑
− 𝑥 + 3𝑦 = 0
Con’t 63 
4𝑥 − 𝑦 = 220
−4𝑥 + 12𝑦 = 0
11y=220
𝑦 = 20
4𝑥 − 20 = 220
4𝑥 = 240
𝑥 = 60
64. Find the exact area of the figure to the left.
1
a. 𝐴 = 2 (3)(4) + 4𝜋
i. 𝐴 = 6 + 4𝜋
1. 𝐴=18.6
65. 120/2 = 60 for <CDE and <ADE therefore it creates a special right triangle 30-60-90
across from 90 is the side length of CD=26
26=2a
a=13 which is across from the 30 degree angle for DE
DE=EB in a rhombus therefore
EB=13 meters
𝑀
66. Arc length of 𝐴𝐵 = 2𝜋𝑟( )
360
120
𝐴𝐵 = 2𝜋(5)(
)
360
𝐴𝐵 = 10.47 𝑐𝑚
67. Area of shaded region is area of sector in the circle
𝑚
𝐴𝐵 = 𝜋𝑟 2 (
)
360
60
𝐴𝐵 = 𝜋(142 )(
)
360
𝐴𝐵 = 102.63 𝑚2
68. Area of shaded region= area of bigger circle-area of smaller circle
Area of circle=𝜋𝑟 2
Smaller circle= 𝜋62
Smaller circle = 36𝜋𝑓𝑡 2
Bigger circle = 𝜋82
Bigger circle= 64𝜋𝑓𝑡 2
64𝜋 − 36𝜋 = 28𝜋𝑓𝑡 2 = 87.96 𝑓𝑡 2
69. (𝑥 − 2)2 +(𝑦 + 3)2 = 16 find the center and radius
𝑐𝑒𝑛𝑡𝑒𝑟 (2, −3)
𝑟𝑎𝑑𝑖𝑢𝑠 = 4 𝑢𝑛𝑖𝑡𝑠
70. 83(2) because its inscribed to get the arc for ABC to be 166
360-166 = 194 for the arc ADC
194-70= 124 for arc DC
124/2=62 because it is inscribed angle = <DCQ
Formulas
1. Areas
a. Angle Measures
i. Interior Angle Measures
1. 180(𝑛 − 2)
ii. Exterior Angle Measures (one)
360
1. 𝑛 𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠
b. Polygons & Figures
i. Parallelograms & Special Parallelograms
1. Area of 
a. Parallelogram: 𝐴 = 𝑏 × ℎ
1
b. Rhombus: 𝐴 = 2 𝑑1 𝑑2
1
c. Trapezoids: 𝐴 = 2 ℎ(𝑏1 + 𝑏2 )
d. Kites: 𝐴 =
𝑑1 ×𝑑2
2
ii. Square
1. 𝐴 = 𝑠 2 𝑂𝑅 𝑠 × 𝑠
iii. Rectangle
1. 𝐴 = 𝑙 × 𝑤
iv. Triangle
1. 𝐴 = 𝑏ℎ
v. Regular Polygons
1
1. 𝐴 = 2 𝑎𝑃
c. Circles
i. Circle
1. 𝐴 = 𝜋𝑟 2
ii. Sector:
𝑚°
1. 𝐴 = 𝜋𝑟 2 (360°)
iii. Segment
𝑚°
1. 𝐴 = 𝜋𝑟 2 (360°) − ⊿𝑏 × ℎ
2. Perimeters & Circumference
a. Square
i. 𝑃 = 4𝑠
b. Rectangle
i. 𝑃 = 2𝑙 + 2𝑤
c. Circle
i. 𝐶 = 2𝜋𝑟 𝑂𝑅 𝜋𝑑
3. Trigonometry
a. Law of Cosines
i. 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
ii. 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
iii. 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
b. Law of Sines
sin 𝑎
sin 𝑏
sin 𝑐
i. 𝑎 = 𝑏 = 𝑐
c. Special Right Triangles
45º
60º
2x
30º
5.
6.
7.
x
x
45º
x 3
4.
x 2
x
i.
Pythagorean Theorem
a. 𝑎2 + 𝑏 2 = 𝑐 2
Distance Formulas
a. Distance
i. 𝑑 = √(𝑥2 |𝑥1 )2 + (𝑦2 |𝑦1 )2
b. Arc Length
𝑛
i. 𝑙 = 360 × 2𝜋𝑟
Mid-Point & Centers
a. Midpoint
(𝑥1 |𝑥2 ) (𝑦1 |𝑦2 )
i.
, 2
2
b. Middle (Mid-segment) of Trapezoid
𝑏 +𝑏
1
i. 1 2 2 𝑂𝑅 2 (𝑡𝑜𝑝 + 𝑏𝑜𝑡𝑡𝑜𝑚)
Circle Theorems
Vocabulary Terms
1. Figures/ Polygons (Names)
a. Decagon: a plane closed figure with ten sides and ten angles
b. Cube: a six-sided solid. All sides are equal squares and all edges are equal.
c. Heptagon: a plane closed figure with seven sides and seven angles.
d. Hexagon: a plane closed figure with six sides and six angles.
e. Nonagon: a plane closed figure with nine sides and nine angles
f. Octagon: a plane closed figure with eight sides and eight angles
g. Parallelogram: a four-sided plane closed figure having opposite sides equal and parallel.
(Opposite angles are equal, and consecutive angles are supplementary.)
h. Pentagon: a five-sided plane closed figure. The sum of its five angles is 540°.
i. Plane figure: shape having only length and width (two dimensional).
j. Polygon: many-sided plane closed figure. Triangle, quadrilateral, pentagon, and so on
k. Quadrilateral: a four-sided plane closed figure. The sum of its four angles equals 360°.
l. Rectangle: a four-sided plane closed figure having opposite sides equal and parallel and
four right angles.
m. Rhombus: a parallelogram with four equal sides.
n. Square: a four-sided plane closed figure having equal sides and four right angles. Its
opposite sides are parallel.
o. Trapezoid: a four-sided plane closed figure with only one pair of parallel sides, called
bases.
p. Triangle: a three-sided plane closed figure. Contains three angles the sum of whose
measures is 180°.
2. Polygons (Theorems)
a. Convex polygon: a polygon in which all diagonals lie within the figure.
b. Concave polygon: a polygon which contains at least one diagonal outside the figure.
c. Diagonal of a polygon: a line segment connecting one vertex to another vertex, and not a
side of the polygon.
d. Median: in a trapezoid, a line segment parallel to the bases and bisecting the legs.
e. Regular polygon: a polygon in which sides and angles are all equal. For example, a regular
pentagon has five equal angles and five equal sides.
f. Apothem: the distance between the center to side of a polygon (perpendicular)
g. Central angle: an angle whose vertex is the center of the figure.
3. Triangles
a. Acute triangle: a triangle containing all acute angles.
b. Equilateral triangle: a triangle in which all three angles are equal in measure and all three
sides have the same length.
c. Hypotenuse: in a right triangle, the side opposite the 90° angle.
d. Isosceles right triangle: a triangle having two equal sides, two equal angles, and one 90°
angle. Its sides are always in the ratio 1, 1, √2.
e. Isosceles triangle: a triangle having two equal sides (and thus two equal angles across
from those sides).
f. Legs: in a right triangle, the two sides forming the 90° angle. In a trapezoid, the nonparallel
sides.
g. Median: in a triangle, a line segment drawn from a vertex to the midpoint of the opposite
side.
h. Obtuse triangle: a triangle containing an obtuse angle
i. Pythagorean theorem: a theorem that applies to right triangles. The sum of the squares of
a right triangle's two legs equals the square of the hypotenuse (a2 + b2 = c2).
j. Right triangle: a triangle containing a 90° angle.
k.
4. Lines
a.
b.
c.
d.
e.
f.
Scalene triangle: a triangle having none of its sides equal (or angles equal).
Intersecting lines: lines that meet at a point.
Line segment: a part of a line; has two endpoints
Parallel lines: two or more lines, always the same distance apart. Parallel lines never meet.
Perpendicular lines: two lines that intersect at right angles.
Ray: a half-line. Continues forever in one direction. Has one endpoint.
Straight line: often described as the shortest distance between two points. Continues
forever in both directions. (Line means straight line.)
5. Angles
a. Acute angle: an angle whose measure is less than 90°.
b. Adjacent angles: angles that share a common side and a common vertex.
c. Angle: formed by two rays with a common endpoint.
d. Complementary angles: two angles the sum of whose measures is 90°.
e. Exterior angle: an angle formed outside the polygon by extending one side. In a triangle,
the measure of an exterior angle equals the sum of the measures of the two remote interior
angles.
f. Corresponding: in the same position. Coinciding.
g. Interior angles: angles formed inside the shape or within two parallel lines
h. Obtuse angle: an angle greater than 90° but less than 180°
i. Right angle: an angle whose measure is equal to 90°.
j. Straight angle: an angle equal to 180°. Often called a line.
k. Supplementary angles: two angles the sum of which measures 180°.
l. Vertical angles: the opposite angles formed by the intersection of two lines. Vertical angles
are equal in measure.
6. Circles
a. Central angle: an angle whose vertex is the center of the circle. The measure of a central
angle is equal to the measure of its arc.
b. Arc: the set of points on a circle that lie in the interior of a central angle
c. Chord: a line segment joining any two points on a circle.
d. Circle: in a plane, the set of points all equidistant from a given point.
e. Circumference: the distance around a circle; equals 2 x π x radius or π x diameter (C = 2πr
or πd.).
f. Concentric circles: circles with the same center.
g. Diameter: a line segment that contains the center and has its endpoints on the circle. Also,
the length of this segment. (A chord through the center of the circle.)
h. Inscribed angle: in a circle, an angle formed by two chords. Its vertex is on the circle. The
measure of an inscribed angle equals one-half the measure of its arc.
i. Radius: a line segment whose endpoints lie one at the center of a circle and one on the
circle. Also, the length of this segment.
j. Tangent to a circle: a line, line segment, or ray that touches a circle at one point (cannot go
within the circle).
7. Miscellaneous Terms
a. Height: altitude. From the highest point, a perpendicular drawn to the base.
b. Area: the space within a shape; measured in square units.
c. Bisects: divides into two equal parts.
d. Congruent: exactly alike. Identical in shape and size.
e. Consecutive: next to each other
f. Midpoint: the halfway point of a line segment, equidistant from each endpoint.
g. Perimeter: the total distance around the outside of any polygon. The total length of all the
sides
h. Pi (π): a constant used in determining a circle's area or circumference. Equals
approximately 3.14 or 22/7
i. Point: a basic element of geometry, a location. If two lines intersect, they do so at a point.
j. Similar: having the same shape but not the same size, in proportion.
k. Surface area: the total surface of all sides of a solid, or the total area of faces.
l. Transversal: a line crossing two or more parallel or nonparallel lines in a plane.
m. Vertex: the point at which two rays meet and form an angle, or the point at which two
sides meet in a polygon.
n. Volume: capacity to hold, measured in cubic units. Volume of rectangular prism = length x
with x height.
Practice Problems
Chapter 10
Chapter 5
Chapter 12