Spring Semester Review Your Spring Assessment has three parts. Part 1: Sometimes, Always, Never The exam contains four SAN questions that you must answer and justify. You will be exposed to all four of these questions in your class SAN activity the week leading up to the exam, so we are not including sample problems here. Part 2: Calculation The exam contains 9 calculation problems spanning our units this semester that incorporate area, similarity, trigonometry, circles, and volume. Some problems are more than one part. The following problems will help you prepare for this calculation section. All answers should be in SRF when possible. If not possible, round to the hundredths. 1. Given that quadrilateral PLUM is a rhombus, with diagonals ML = 12 and PU = 16, find !αPLUM . 2. Given the isosceles trapezoid, PEAR, with bases PE=12 cm, AR=28 cm and area 300 cm2. a. Find the height of the trapezoid b. Find the length PA. 3. Given a = 14, b = 16, c = 21, classify triangle ABC by sides and angles. Explain your reasoning. 4. Draw right triangle ABC so that !!A is a right angle and !AD is the altitude of ABC so that B-­‐D-­‐C. Led BD = 9 and CD = 16. a. Find AD. b. How is your calculation from a) related to the geometric mean? c. Find AB and AC. 5. Find the exact area and perimeter of triangle ABC below. !BD is an altitude of triangle ABC. B !α △ABC = _______ 5 2 perim!△ABC = _______ 45°
A
D
12
C
6. Find the area and perimeter of trapezoid MNOP. !αMNOP = ______ 12
perim !MNOP = ______ 7
10
7. In tI he diagram below, KL = 8, JK = x, JI = 2, LM = 6, MN = 3, NO = y. I-­‐J-­‐K-­‐L, O-­‐N-­‐M-­‐L Solve for x and y. J
x = ______ K
L
M
N
O
y = ______ 8. Given below: RP is the angle bisector of !!QPS , QR = x, RS = 9, SP = 10, QP = 13, Q-­‐R-­‐S. Find x. S
x = ______ R
Q
P
9. Given the marked figure below (you may assume collinearity), solve for x and y. y
45°
x= __________ x
3
y= __________ 10. Given: U-­‐Y-­‐X, T-­‐Y-­‐W-­‐V, !TV ⊥ UX,UT ⊥ TX,!TV!is!an!angle!bisector!of!!UTX,!m!YWX=30" ,!YW = 4 3;!WV = 6 Solve for all missing values. Write them to the right of the figure below. Round any angles to the hundredth if necessary. All sides should be in SRF. U
T
Y
V
W
X
11. Given a = 12, !m!A = 38" ,m!Y = 62" , W is the midpoint of YV . Sketch and solve triangle YWA . ( )
12. In rhombus ABCD, cos !A =
!
3
. If AB = 12, find the area of the rhombus. 2
!#"
13. Given the radius of circle A is 6 and BC = 8, and !BC is a tangent line, A-­‐D-­‐C, find DC. DC = _______ B C
D
A
14. Quadrilateral FUNK is inscribed in circle Y with !m!F = 75" ,m!U = 80" . Draw the figure and find "
!m!N!and!mFUN . m!N = _____
"
!mFUN = _____
!
#
!
15. Given: A-­‐Y-­‐R, F-­‐Y-­‐D, A-­‐D-­‐I, F-­‐R-­‐I !mFR = 100" ,mAD = 110" ,mFA = 105" . Find: A
m!FYR = _____
F
Y
D
R
m!RAD = _____ !m!FIA = ______
I
16. Find the area of a sector of a circle with radius 12 if the arc length of the sector is 18π 17. Find the area between a circle of radius 8 and an inscribed octagon. 18. Find the total area of a regular hexagonal pyramid with an edge length of 15 and a base radius of 12. 19. Two non-­‐intersecting circles with radii of 10” and 8” have centers that are 24” apart. Sketch a diagram and find the area bounded by the circles and their common external tangents. 20. Consider the diagram of !⊙M where !PT is a tangent segment, T, A, B on the circle and P-­‐A-­‐B. !
a. If !m!TMA = 80 , find !mTBA . b. If!m!M = 80 and !m!P = 50 , find !m!MAP c. If PA = 9 and AB = 16, then PT = ? T
M
B
A
P
!
d. If !PS is drawn tangent to circle M at S, and !m!SPT = 62 , then !mST = ? 21. Find the area of a square with the radius of its circumcircle measuring 6. 22. Given GHIF is a rectangle and L and K are the centers of semicircles, Find the shaded area. G
L
J
K
F
23. The surface area of a right, rectangular prism with dimensions of 8, 10, and 16 is… 24. An isosceles trapezoid has a height of 9 and an area of 117. If one base is ½ the length of the other, find both bases. 25. If the perimeter of an equilateral triangle is 24, its area is… H
M
I
26. The area of a triangle with sides 15, 15, and 24 is… 27. The area of a regular hexagon inscribed in a circle of radius 8 is… 28. If sinθ =
!
3
ad !cosθ < 0 , then !tanθ = ?? 5
29. If the long leg of a 30-­‐60-­‐90 triangle is !7 2 , then the hypotenuse is… 30. Classify a triangle with side lengths 12, 13, and 18. Find the measures of its angles. 31. Triangle ABC has dimensions a = 6, b = 9, and !m!A = 40 . Solve for BOTH possible triangles with these dimensions. 32. The surface area of a rectangular prism with dimensions 12, 6, and 9 is.. 33. A cube is inscribed in a cylinder with radius 5. The volume of the cube is… 34. Find the area and perimeter of an octagon if the incircle radius is 10. 35. Consider the diagram of parallelogram ACDG so that A-­‐B-­‐C and D-­‐E-­‐F-­‐G α △FBE
= !α △ACF
α CDEB
b. Find !α ACDG
α △FBE
c. Find !α ACDG
α GBE
d. Find !α ABFG
D
C
a. Find E
B
F
A
G
36. Let cosθ =
!
3
and !180 < θ < 360 . Find !sinθ and !tanθ 8
Part 2: Proofs There are two proofs on the final assessment and you must choose one. Here are some proofs to practice: 1. If two quadrilaterals are similar, then the ratio of any two corresponding sides is equal to the ratio between corresponding diagonals. 2. Let TONG be a quadrilateral and let S be the intersection of the diagonals. If (TS)(GS)=(OS)(NS) where !TS ≠ SN , prove that TONG is a trapezoid. 3. In the diagram below of Circle A and square BCDE (so R-­‐B-­‐G, G-­‐C-­‐H, F-­‐E-­‐D-­‐A-­‐H, and H, F, G on circle A), prove that DE is the geometric mean of HD and FE.