Geometry L2 Name: _________________________________ Midterm Exam Review The midterm exam will cover the topics listed below from Units 1, 2 and 3. A formula sheet will be provided. Unit 1 β Transformations Pythagorean Theorem Simplifying radicals and solving application problems Distance and Midpoint on the Coordinate Plane
Distance Formula: π = β(π₯2 β π₯1)2 + (π¦2 β π¦1)2
Midpoint Formula π (π₯1+π₯2
2,
π¦1+π¦2
2)
Partition a Segment Find a partition point given a fraction, ratio, or percentage Isometries Translations Using Mapping Notation and Vector Notation
Naming Vectors, Component Form, Length of a Vector: βπ2 + π2 Reflections Over x-axis, y-axis, y = x, y = -x Over vertical lines (x = ____); Over horizontal lines (y = ____) Rotations 90Β°, 180Β°, 270Β° about origin on coordinate plane Composition of Transformations on the coordinate plane Dilations β Graph a figure and its dilation on the coordinate plane Symmetry - Determine Line Symmetry and Rotational Symmetry of figures Unit 2 β Congruence and Proof Angles Formed by Parallel Lines and Transversals Corresponding, Alternate Interior, Alternate Exterior, Consecutive (Same-side) Interior Vertical Angles, Linear Pairs, Complementary and Supplementary Angles Triangle Sum Theorem Exterior Angle Theorem Congruent Triangles/Congruent Polygons - Corresponding Parts
Proving Triangles Congruent β SSS, SAS, ASA, AAS, HL CPCTC Two-Column Proofs involving congruent triangles and CPCTC Isosceles and Equilateral Triangles β Properties and Theorems
Unit 3 - Polygons Polygon Angle Sum Theorems Interior Angle Sum: (n β 2) 180Β° Exterior Angle Sum: 360Β°
Regular Polygons: Each Interior Angle: (πβ2)180Β°
π
Each Exterior Angle: 360Β°
π
The best way to prepare for your Geometry Midyear Exam is to complete the problems in this review packet and to study your tests and quizzes from units 1, 2, and 3. I. Use the Pythagorean Theorem to find the length of the missing side in each triangle. Write answers in simplest radical form. 1. 2.
x=___________ x=___________ 3. 4.
x=___________ x = ___________
8
5
x
12
β5
β11
x
x 4β3
8
x 12
5. Find the height and the area of a rectangle which has a diagonal of length 26 and a base of length 24. Height = ________ Area = __________ 6. An isosceles triangle has legs of length 34 and a base of length 32. Find its height and its area.
height = __________ Area = ____________ II. Distance and Midpoint Formulas 7. On the coordinate plane, π πΜ Μ Μ Μ has coordinates R (2, - 7) and S (-4, 1).
a. Find the length of . b. Find the midpoint (M) of .
c. Use the distance formula to prove that M is the midpoint of . 8. M is the midpoint of ππΜ Μ Μ Μ . If X has coordinates (-3, 5) and M has coordinates (1, 0), find the coordinates of point Y.
RS RS
RS
III. Partition A Segment 9. Point A has coordinates (-3, 2). Point B has coordinates (3, -7).
Point C is located 2/3 of the way from A to B. Find the coordinates of point C.
10. Point G has coordinates (4, 3). Point H has coordinates (-1, -7).
Point J divides GH in a 1:4 ratio. Find the coordinates of point J. IV. Transformations 11. Use the translation (x, y) β (x + 3, y β 4):
a. What is the image of D (4, 7)? b. What is the pre-image of Mβ (-5, 3)?
12. The vertices of Ξ MNO are M (-2, 4), N (-1, 1), and O (3, 3). Graph Ξ MNO and its image using prime notation after
the translation (x, y) β (x + 4, y β 2):
Mβ: ________ Nβ: ________ Oβ: ________
13. ΞRβSβTβ is the image of ΞRST after a translation. Write a rule for the translation in mapping notation and in vector
notation. Mapping Notation: Vector Notation:
14. Name the vector, write its component form, and find its length:
a. b.
15. Write the component form of the vector that describes the translation from S (-3, 2) to Sβ (6, -4). 16. The vertices of ΞABC are A (0, 4), B (2, 1) and C (4, 3). Graph and label the coordinates of ΞAβBβCβ after each
transformation. a. Translate ΞABC using the vector β¨β3, 1β©. b. Reflect ΞABC over the x-axis.
R
S
T Sβ
Tβ
Rβ
c. Reflect ΞABC over the line y = - x. d. Reflect ΞABC over the line y = -1.
17. The vertices of ΞABC are A (-3, 1), B (1, 1) and C (1, -2). Reflect ΞABC over the line x = 2. Then reflect ΞAβBβCβ over
the line y = -3. Graph ΞABC, ΞAβBβCβ, and ΞAβBβCβ. State the coordinates of ΞAβBβCβ.
Aβ:________ Bβ:________ Cβ:________
18. The coordinates of βABC are A (0, 4), B (3, 6), and C (5, 2). Graph βABC. Rotate βABC 90Β°, 180Β°, and 270Β° counterclockwise about the origin. Record the coordinates after each rotation.
After a 90Β° Rotation: Aβ ________ Bβ _______ Cβ ________
After a 180Β° Rotation: Aβ ________ Bβ _______ Cβ ________
After a 270Β° Rotation: Aβ ________ Bβ _______ Cβ ________
19. List the image of each of the following points after the specified composition of transformations:
a. If point A (-2, 5) is reflected in the y-axis,
and then point Aβ is reflected in the x-axis, the coordinates of point Aββ are _________.
b. If point B (-4, -2) is reflected over the line y =- x, and then point Bβ is rotated 90Β° counterclockwise about the origin, the coordinates of Bββ are _________.
c. If point C (6, -3) is reflected over the line y = x, and then point Cβ is rotated 270Β° counterclockwise about the origin, the coordinates of Cββ are _________.
d. If point D (-2, 10) is rotated 180Β° about the origin, and then point Dβ is reflected over the line y=-5, then the coordinates of Dββ are ________.
20. Point P (-6, 2) is transformed to point Pβ (2, 6). What is the transformation that maps P into Pβ? Explain. 21. The vertices of βABC are A (2, 4), B (7, 6) and C (5, 2). Graph the image of βABC after a composition of the
transformations in the order they are listed.
Transformation: (π₯, π¦) β (π₯ + 2, π¦ β 4) Rotation: 180Β° about the origin Record the coordinates of βAβBβCβ: Aβ _________ Bβ __________ Cβ __________
22. The coordinates of βPQR are: P (6, 2), Q (-6, 4), R (0, -4).
a. Draw the dilation of βPQR centered at the origin with a scale factor of 2
Pβ ________ Qβ _________ Rβ __________
b. Draw the dilation of βPQR centered at the origin with a scale factor of Β½
Pβ ________ Qβ _________ Rβ __________
V. Symmetry 23. State the number of lines of symmetry and the angle(s) of rotational symmetry of each of the following figures: a. Rectangle b. Isosceles Triangle c. Square # of Lines:_______ # of Lines:_______ # of Lines:_______ Angle(s):____________ Angle(s):_____________ Angle(s):_____________ d. Regular Hexagon e. Equilateral Triangle f. Parallelogram # of Lines:_______ # of Lines:_______ # of Lines:_______ Angle(s):____________ Angle(s):_____________ Angle(s):_____________
4
2
3
5
m
n
6 7
8
125Β°
x 35Β° 85Β°
y
33Β° xΒ°
yΒ° 25Β°
23Β°
VI. Parallel Lines Intersected by a Transversal 24. Name the following pairs of angles in the diagram below:
Corresponding Angles (4 pairs) β
Alternate Interior Angles (2 pairs) β
Alternate Exterior Angles (2 pairs) β
Consecutive (Same β Side) Interior Angles (2 pairs) -
Vertical Angles (4 pairs) β
Linear Pairs (name 4 of the 8 in the diagram) β
25. In the figure above, if line m is parallel to line n, and the measure of angle 1 is 106Β°, find the measures of the other seven angles:
m < 2 = ______ m < 3 = ________ m < 4 = ________ m < 5 = ________
m < 6 = ______ m < 7 = ________ m < 8 = ________
VII. Angles of Triangles
Find the missing angle(s) in each of the following figures:
26. 27.
28. 29.
1
A
D B C
C
5xΒ°
7x+42Β° 18xΒ°
y
115Β° 135Β°
x
y
30. 31.
32. 33.
34. Angle DBA is an exterior angle of βABC. Find the measure of angle ABC.
m < ABC = ________
35. In ΞDEF, m < D = 7x + 10Β°, m < E = 9x -1Β°, and m < F = 3x + 38Β°. Find the measures of the angles of the triangle. What type of triangle is βDEF?
m < D = _______
m < E = _______
m < F = _______
Triangle Type: _________________
A
B C
P
Q R
90Β°
23Β°
12
5
VIII. Congruent Triangles
36. βπ΄π΅πΆ β βπππ m < P = _______ m < Q = _______ PQ = ________ QR = ________
Determine the theorem (if any) which can be used to prove the triangles congruent.
37. ___________ 38. ___________ 39. ____________
40. ___________ 41. ___________ 42. ___________
Determine the third congruence which is needed to prove the triangles congruent by the indicated method.
43. SAS 44. ASA 45. AAS 46. HL
A
C B
E F
D
Statements Reasons
Statements Reasons
Statements Reasons
IX. Two β Column Proofs
47. Given: π΄π΅Μ Μ Μ Μ β π΄π·Μ Μ Μ Μ
πΆ ππ π‘βπ ππππππππ‘ ππ π΅π·Μ Μ Μ Μ
Prove: βπ΄π΅πΆ β βπ΄π·πΆ
48. Given: ππΜ Μ Μ Μ Μ πππ ππΜ Μ Μ Μ πππ πππ‘ πππβ ππ‘βππ
Prove: βπππ β βπππ
49. Given: π·π΅Μ Μ Μ Μ β₯ π΄πΆΜ Μ Μ Μ
π΄π΅Μ Μ Μ Μ β π΅πΆΜ Μ Μ Μ
Prove: < π΄ β < πΆ
Statements Reasons
N
T
M
Q
x
50. Given: ππΜ Μ Μ Μ β ππΜ Μ Μ Μ Μ ππΜ Μ Μ Μ || ππΜ Μ Μ Μ Μ Prove: < π β < π
X. Isosceles and Equilateral Triangles
51. 52. x = ______
y = ______
π < 1 = ____ π < 2 = ______
π < 3 = ______ π < 4 = ________
53. 54.
a= _______ x = ________ b = ________
c = _______ d = ________
130Β°
y x
A
B C
S
Q R
R Q
S
55. In ΞABC, m < A = 4x + 22Β°, m < B = 12x - 2Β°, and m < C = 8x + 16Β°. Is ΞABC isosceles? Explain why or why not.
56. ΞQRS is isosceles with base ππ Μ Μ Μ Μ . If SQ = 14x β 3 cm, SR = 8x + 9 cm, and QR = 5x + 13 cm, find the perimeter of the triangle.
Perimeter = ____________
57. ΞQRS is isosceles with base ππ Μ Μ Μ Μ . If m < Q = (6x + 3)Β° and m < R = (3x +21)Β°, find the measures of the angles of the triangle.
m < Q = _______
m < R = _______
m < S = _______
XI. Angles of Polygons
58. Find the interior angle sum and the exterior angle sum of a 15-gon.
Interior Angle Sum = ________
Exterior Angle Sum = _______
59. A polygon has an interior angle sum of 2880Β°. How many sides does it have?
60. Find the measure of each interior angle and each exterior angle of a regular nonagon.
Each Interior Angle = _________
Each Exterior Angle = _________
61. Each exterior angle of a regular polygon has a measure of 30Β°. Name the polygon.
62. Each interior angle of a regular polygon has a measure of 135Β°. Name the polygon.
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