Download 00-Spring semester review

Geometry GT/Pre-AP Spring Semester Review
Name: ___________________________
Date: ________________ Period: _____
Textbook chapters 8, 6, 10, 11, 12
Due May 26 or 27:
Due May 28 or 29:
Exercises for Units 7-8, 9, 10:
Exercises for Units 11, 13, 12:
Right Triangles & Trig, Quadrilaterals, Area.
Surface Area & Volume, Pre-AP & SAT, Circles.
This review counts as a TWO DAILY GRADES. Show work on a separate sheet. Write answers in blanks provided.
Units 7 & 8: Right Triangles and Trigonometry (chapter 8)



Apply the Pythagorean Theorem and Pythagorean Triples to find unknown sides.
o a2 + b2 = c2
or leg2 + leg2 = hyp2
o Some triples are 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41
o You will often be asked to solve a multi-step problem which requires Pythagorean Theorem or triples
but then asks for the perimeter, area, etc.
Determine whether a triangle is a right, acute, or obtuse triangle.
o Compare “c” (the largest side) with the other two sides.
o If c2 = a2 + b2 then it is a right triangle.
o If c2 < a2 + b2 then it is an acute triangle (all angles are < 90°).
o If c2 > a2 + b2 then it is an obtuse triangle (one angle is > 90°).
Identify and apply special right triangle relationships to find lengths of unknown sides.
30°
s



60°
s
3
90°
2s
OR
45°
45°
s
s
90°
s
2
Simplify radicals, add/subtract/multiply/divide radicals, and rationalize the denominator.
Use trigonometric ratios to find unknown sides or angles of right triangles.
o SOH CAH TOA:
opposite
adjacent
opposite
cos 
sin  
tan  
hypotenuse
hypotenuse
adjacent
angle between
the ladder and
the building
Angle of
elevation
o To find an unknown side, set up the equation and solve as a proportion.
o To find an unknown angle, press 2ND SIN or 2ND COS or 2ND TAN to get the inverse operation.
Find the angle of elevation or angle of depression, or use them to find unknown sides
o Angle of elevation/depression is always starting at the HORIZONTAL and angling up or down.
o An exception is if the problem asks for the angle “from the vertical” or “the angle between the ladder
and the building.”
Exercises for Units 7 & 8 (chapter 8): Right Triangles and Trigonometry
The lengths of three sides of a triangle are given. Describe each triangle as acute, right, or obtuse.
1. __________ 14, 48, 50
2. __________ 6, 7, 9
Find the values of the variables. Leave your answers in simplest radical form.
3. x = __________
4. x = __________
y = __________
5. x = __________
y = __________
What are the measures of
the two congruent angles?
Continue to next page…
Find the value of x. Round lengths of segments to the nearest tenth and angle measures to the nearest degree.
6. __________
7. __________
8. __________ A surveyor measures the top of a building 50 ft away from him. His angle-measuring device is 4 ft
above ground. The angle of elevation to the top of the building is 63°. How tall is the building?
9. __________ A forest ranger looking out from a ranger’s station can see a forest fire at a 35° angle of
depression. The ranger’s position is 100 ft above the ground. How far is it from the ranger’s station to the fire?
10. __________ A moving van traveled 200 mi west, then 70 mi south, then 50 mi east, and finally 100 mi north.
Find the distance from the point of origin to the destination (to the nearest mile), and the direction traveled.
Find the value of each variable. Leave your answers in simplest radical form. Then find the perimeter and area of
each composite figure.
11. w = __________
x = __________
y = __________
z = __________
perimeter = ____________________
12. p = __________
q = __________
r = __________
s = __________
perimeter = ____________________
Unit 9: Quadrilaterals (chapter 6)

Identify special quadrilaterals and use their properties to solve problems.
o Parallelogram
 Both pairs of opposite sides parallel and congruent ()
 Both pairs of opposite angles congruent, and consecutive angles supplementary
 Diagonals bisect each other
o Rectangle
 All the properties of a parallelogram
 4 right angles
 Diagonals are congruent
o Rhombus
 All the properties of a parallelogram
 4 congruent sides
 Diagonals are perpendicular, and diagonals bisect opposite angles
o Square
 All the properties of a parallelogram AND rectangle AND rhombus!
o Trapezoid
 Exactly 1 pair of opposite sides parallel
 The midsegment is parallel to the bases and is the average (or ½ the sum) of the bases
o Isosceles Trapezoid
 All the properties of a trapezoid
 Legs are congruent, and base angles are congruent
 Diagonals are congruent
o Kite
 Exactly 2 pairs of consecutive sides congruent
 Exactly 1 pair of opposite angles congruent (the “elbows”)
 Diagonals are perpendicular

Prove a quadrilateral is a special quadrilateral.
o On a coordinate plane, plot the 4 vertices and
 Use slope (rise/run) to prove parallel (same slope) or right angles (opposite reciprocals)
 Use Pythagorean Theorem or Distance Formula to prove congruent sides or diagonals
o Parallelogram: By any of the 5 properties above, OR by showing 1 pair of opposite sides both parallel
and congruent
o Rectangle: It is a parallelogram AND… Congruent diagonals OR 1 right angle
o Rhombus: It is a parallelogram AND… 1 pair congruent consecutive sides OR perpendicular diagonals
OR 1 diagonal bisects opposite angles
o Square: It is a parallelogram AND a rectangle AND a rhombus!
Exercises for Unit 9 (chapter 6): Quadrilaterals
Graph quadrilateral ABCD. Then determine the most precise name for each quadrilateral.
Then find the perimeter of each quadrilateral. Leave your answers in simplest radical form.
13. Name __________
Perimeter __________
A(2, 3), B(-4, 3), C(-2, 6), D(1, 6)
14. Name __________
Perimeter __________
A(0, 6), B(3, 3), C(0, -5), D(-3, 3)
15. Name __________
Perimeter __________
E(0, 4), F(3, 0), G(7, 3), H(4, 7)
16. __________ QUVX is a rectangle with Q(-7, -3) and Z(-2, 1). What are the coordinates of U?
17. Parallelogram
x = __________
y = __________
18. Parallelogram
x = __________
19. Rhombus
1 = _____, 2 = _____
perimeter = __________
Give the coordinates of the missing points without using any new variables.
20. Q (_____, _____)
Parallelogram
21. D(_____, _____), E(_____, _____) 22. A(____, 4b), L(____, ____), T(0 ,____)
Isosceles Trapezoid
Rhombus
Unit 10: Area (chapter 10)



Calculate the areas of the following shapes.
o A=bh
for a parallelogram or a rectangle, where the base and height form a right angle!
o A = s2
for a square
o A=½bh
for a triangle
o A = ½(b1 + b2)h for a trapezoid
o A = ½ d 1 d2
for a rhombus or a kite, where d = diagonal
o A=½ap
for a regular polygon, where a = apothem, and p = perimeter
o Composite shapes
Find arc lengths and sector areas of circles.
o
o Circumference = 2r
Area =  r2
Use ratios of perimeter and area to solve similar figures.
o If the similarity ratio is a : b, then the perimeter ratio is a : b, and the area ratio is a 2 : b2
o Prices for a 2-dimensional figure (like a field, carpet, table mat, etc.) use the area ratio
Exercises for Unit 10 (chapter 10): Area
If your answer is not an integer, round to the nearest tenth.
23. __________ Find the area of an equilateral triangle with side length of 6 ft.
24. __________ regular hexagon with side length of 4 cm.
25. __________ Find the area of an isosceles triangle with legs each 20 ft long and a base 24 ft long.
26. Area = __________
27. Length of arc PQ = __________
Area of sector PNQ = __________
28. ________% Probability that a
random point is in shaded region
in terms of 
29. __________ Benita plants the same crop in two rectangular fields, each with side lengths in a ratio of 2 : 3.
Each dimension of the larger field is 3.5 times the dimension of the smaller field. Seeding the smaller field costs
$8. How much money does seeding the larger field cost?
Find the area of the figures from the following questions above.
30. __________ the quadrilateral in question 13
31. __________ the quadrilateral in question 14
32. __________ the quadrilateral in question 15
33. __________ the parallelogram in question 20
34. __________ the triangle in question 11
35. __________ the triangle in question 12
Unit 11: Surface Area and Volume (chapter 11)

Calculate the lateral area, total surface area, and volume of the following 3-D figures.
o Prisms and Cylinders
 LA = Ph
where P = perimeter of the base.
 SA = LA + 2B
h = distance between the two bases.
 V = Bh
B = area of the base, which depends on the SHAPE of the base.
o
Pyramids and Cones
 LA = 12 P ℓ


o
SA = LA + B
V = 13 B h
or
LA =  r ℓ
for cones.
where ℓ = slant height along a lateral face.
h = perpendicular height from the base to the vertex.
Sphere
 SA = 4  r2
 V = 43  r3
 A hemisphere is half a sphere.
Composite figures: be careful when calculating surface area – do not include a base area if it lies
inside the figure. Break it down into parts, for example: ½ SA of sphere + LA of cylinder + 1 base.
o Oblique figures use the same formulas as right figures.
Use ratios of similarity, area, and volume to solve similar solids.
o If the similarity ratio is a : b, then the area ratio is a2 : b2, and the volume ratio is a3 : b3.
o Weight uses the volume ratio.
o If given 2 lengths, areas, or volumes… first SIMPLIFY! (Divide in calculator and press MATH 1:Frac).
o Then find the similarity ratio, if not already given.
 If given 2 areas, take the square root of both numbers.
 If given 2 volumes, take the cube root (press MATH 4: 3√ ) of both numbers.
o Then get the ratio of what you want to find.
 If want to find area, then square both numbers.
 If want to find volume, then cube both numbers (press MATH 3: 3 ).
o Finally, set up a proportion with the ratio and the known value.
o

Exercises for Unit 11 (chapter 11): Surface Area and Volume
Find the surface area (SA) and volume (V) of each figure to the nearest tenth.
36. SA = __________
V = __________
37. SA = __________
V = __________
38. SA = __________
V = __________
39. SA = __________
V = __________
40. __________ Two similar cones have heights of 9 cm and 4 cm. Find the ratio of their volumes.
41. a) __________ Two cylinders are similar, with surface areas of 25  cm2 and 49 cm2. What is the similarity
ratio?
b) __________ If the smaller cylinder has a volume of 50 cm3, what is the volume of the larger cylinder?
42. a) __________ What space figures can you use to approximate the shape of the ice cream cone?
__________
b) __________ Find the entire figure’s surface area to the nearest tenth.
c) __________ Find the entire figure’s volume to the nearest tenth.
43. __________ The “chocolate blast” ice cream cone has a spherical chocolate candy center
of radius 3 cm, thus decreasing the amount of ice cream in the cone. What
is the new volume of ice cream that the ice cream cone can hold?
44. __________ All the ice cream in a cylindrical carton (with radius 5 cm and height 10 cm) is
used to make smaller ice cream cones of 32.5 cm3. How many can be made?
Unit 13: Pre-AP and SAT Topics






Write equations of piecewise linear functions.
Find the area under a curve bounded by the x-axis in a given interval.
Write transformations (translations and reflections) in function notation.
Describe the effect of transformations on slope and segment lengths.
Approximate area under a curve by finding the left sum, right sum, and trapezoid sum.
Calculate the geometric probability of lengths and areas (10-8 in textbook).
o Find the favorable region, and divide by the total region.
o For bus or traffic light cycles, draw a number line. Find the point on the line where the wait is exactly
the given minutes. Then decide whether to shade to the left (wait > given) or to the right (wait <
given). Find the shaded length, and divide by the total length of the number line.
Exercises for Unit 13: Pre-AP and SAT Topics
Refer to the function f(x) on the right. On what open intervals is…
45. a) ___________________________________ f(x) increasing?
b) ___________________________________ f(x) decreasing?
c) ___________________________________ f(x) constant?
46. a) ___________________________________ slope of f(x) increasing?
b) ___________________________________ slope of f(x) decreasing?
c) ___________________________________ slope of f(x) constant?
Find the:
A
B
C
D
E
O
… of f(x) for
… of f(x) for
… of f(x) for
… of f(x) for
… of f(x) for
0<x<1 ( OA )
1<x<3 ( AB )
3<x<7 ( BC )
7<x<9 ( CD )
9<x<10 ( DE )
47. Slope
(rise/run)
48. Equation
in slopeintercept form
49. Length
(distance formula
or draw a right ∆)
50. Area under
curve (rectangle,
triangle, trapezoid)
Write an equation for a new function g(x) in terms of f(x) that will…
51. a) _______________ translate f 5 units up?
b) _______________ translate f 5 units down?
c) _______________ translate f 5 units right?
d) _______________ translate f 5 units left?
e) _______________ reflect f across the y-axis?
f) _______________ reflect f across the x-axis?
52. a) _______________ How are the slopes of a function affected by a translation?
b) _______________ How are the segment lengths of a function affected by a translation?
c) _______________ How are the slopes of a function affected by a reflection?
d) _______________ How are the segment lengths of a function affected by a reflection?
53. __________ An archery target with a radius of 61 cm has 5 scoring zones formed
by concentric circles. The colors of the zones are yellow, red, blue, black, and white.
The radius of the yellow circle is 12.2 cm. The width of each ring is also 12.2 cm. If an
arrow hits the target at a random point, what is the probability that it hits the black
zone (second zone from the outside)?
Unit 12: Circles (chapter 12)


Solve for missing lengths or angles using:
o Properties of tangents
o Properties of chords and arcs
o Properties of inscribed angles and central angles
o Relationships between secants and tangents
o All these properties, relationships, etc. can be found on the “Key Ideas for Circles – Chapter 12”
Write the standard equation for a circle in the coordinate plane
o Circle equation is
(x – h)2 + (y – k)2 = r2
where (h, k) is the center and r is the radius.
Exercises for Unit 12: Circles
Find the measure of arc AB.
54. arc AB = __________
55. arc AB = __________
56. a) _______ b) _______ c) _______ d) _______
Find the value of the variable(s). Assume that lines that appear to be tangent are tangent. Round to the tenth.
57. __________
58. __________
59. __________
60. __________
61. x = ______ y= ______
62. __________
63. __________
64. __________
65. __________ Write an equation of the circle that passes through (2, 8) with center (-3, 4).
66. x = __________ SW = __________ Find x and SW in simplest radical form if YW = 5a – 24, and TW = 3a.
Math web sites that may help you as you review for your 2nd semester exam:
www.purplemath.com, www.coolmath.com (there is a graphing calculator on http://www.coolmath.com/graphit/
), www.math.com, www.themathwebsite.com, and of course your textbook, Pearson Prentice Hall Geometry,
online. This site offers homework video tutors, lesson quizzes, chapter tests, vocabulary quizzes, and real-world
applications. The quizzes and tests for chapters 8, 6, 10, 11, 12 will be especially helpful in testing yourself to
prepare for the final – and its multiple choice!
http://www.phschool.com/
Enter web code “ aue 0775 ” and click Go.
… and REVIEW ALL SPRING SEMESTER OLD TESTS!!!
Exercises for Units 7 & 8 (chapter 8):
1. right.
2. acute.
3. x = 3.
4. x = 7 2 , y = 7 2 .
Exercises for Unit 11 (chapter 11):
36. SA = 360 cm2, V = 400 cm3.
37. SA = 452.4 cm2, V = 904.8 cm3.
38. SA = 603.2 m2, V = 1206.4 m3.
39. SA = 3327.7 cm2, V = 12,250.6 cm3.
40. 729 : 64.
41. a) 5: 7. b) 137.2 cm3 ≈ 431.0 cm3.
42. a) a cone and half of a sphere. b) SA = 251.3
cm2. c) V = 323.6 cm3.
43. 146.6 cm3.
44. 24 ice cream cones.
5. x = 4, y = 4 3 .
6. 67.
7. 20.9.
8. 102 ft.
9. 143 ft.
10. 153 miles northwest.
11. w =
10 3
3
, x = 5, y = 5 2 , z =
5 3
3
,
perimeter = 5 + 5 2 + 5 3 .
12. p = 4 3 , q = 4 3 , r = 8, s = 4 6 ,
perimeter = 12 + 4 3 + 4 6 .
Exercises for Unit 9 (chapter 6):
13. Trapezoid, perimeter = 9 + 10 + 13 units.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Kite, perimeter = 6 2 +2 73 .
Square, perimeter = 20 units.
(3, 5).
x = 2, y = 5.
x = 20.
1 = 53°, 2 = 37°, perimeter = 40 units.
Q (x + k, m)
D (-c, 0), E (0, b)
A (a, 4b), L (2a, 2b), T (0, 2b)
Exercises for Unit 10 (chapter 10):
23. 15.6 ft2.
24. 41.6 cm2.
25. 192 ft2.
26. 120 units2.
27. arc length 1.2 ≈ 3.8cm, sector 1.8 ≈ 5.7cm2.
28. 21.5%.
29. $98.
30. (#13) 13.5 units2.
31. (#14) 33 units2.
32. (#15) 25 units2.
33. (#20) Area = x∙m.
34. (#11) 19.7 units2.
35. (#12) 37.9 units2.
Exercises for Unit 13:
45. a) 0<x<1 and 1<x<3. b) 3<x<7 and
9<x<10. c) 7<x<9.
46. a) none. b) none. c) all of them: 0<x<1,
1<x<3, 3<x<7, 7<x<9, and 9<x<10.
47. 3, 12 ,  12 , 0, –2.
48. y  3x , y 
1
2
x  2.5 , y   12 x  5.5 , y  2 ,
y  2x  20 .
49. 10 , 5 , 2 5 , 2, 5 .
50. 1.5 u2, 7 u2, 12 u2, 4 u2, 1 u2.
51. a) g(x) = f(x)+5. b) g(x) = f(x)–5. c) g(x) =
f(x–5). d) g(x) = f(x+5). e) g(x) = f(–x). f) g(x)
= –f(x).
52. a) same. b) same. c) opposites. d) same.
53. 28%.
Exercises for Unit 12 (chapter 12):
54. 120.
55. 105.
56. a) 170. b) 85. c) 10. d) 85.
57. 40.
58. 115.
59. 80.
60. 44.
61. x = 45, y = 15.
62. 5.5.
63. 6.
64. 11.8.
65. (x + 3)2 + (y – 4)2 = 41.
66. x = 4 3 , SW = 36 + 4 3 .
GOOD LUCK!!!