geometry l2 name:
TRANSCRIPT
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 โ Transformations Pythagorean Theorem Simplifying radicals and solving application problems Distance and Midpoint on the Coordinate Plane
Distance Formula: ๐ = โ(๐ฅ2 โ ๐ฅ1)2 + (๐ฆ2 โ ๐ฆ1)2
Midpoint Formula ๐ (๐ฅ1+๐ฅ2
2,
๐ฆ1+๐ฆ2
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: โ๐2 + ๐2 Reflections Over x-axis, y-axis, y = x, y = -x Over vertical lines (x = ____); Over horizontal lines (y = ____) Rotations 90ยฐ, 180ยฐ, 270ยฐ about origin on coordinate plane Composition of Transformations on the coordinate plane Dilations โ Graph a figure and its dilation on the coordinate plane Symmetry - Determine Line Symmetry and Rotational Symmetry of figures Unit 2 โ 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 โ SSS, SAS, ASA, AAS, HL CPCTC Two-Column Proofs involving congruent triangles and CPCTC Isosceles and Equilateral Triangles โ Properties and Theorems
Unit 3 - Polygons Polygon Angle Sum Theorems Interior Angle Sum: (n โ 2) 180ยฐ Exterior Angle Sum: 360ยฐ
Regular Polygons: Each Interior Angle: (๐โ2)180ยฐ
๐
Each Exterior Angle: 360ยฐ
๐
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
โ5
โ11
x
x 4โ3
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, ๐ ๐ฬ ฬ ฬ ฬ 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 ๐๐ฬ ฬ ฬ ฬ . 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) โ (x + 3, y โ 4):
a. What is the image of D (4, 7)? b. What is the pre-image of Mโ (-5, 3)?
12. The vertices of ฮ MNO are M (-2, 4), N (-1, 1), and O (3, 3). Graph ฮ MNO and its image using prime notation after
the translation (x, y) โ (x + 4, y โ 2):
Mโ: ________ Nโ: ________ Oโ: ________
13. ฮRโSโTโ is the image of ฮRST 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 Sโ (6, -4). 16. The vertices of ฮABC are A (0, 4), B (2, 1) and C (4, 3). Graph and label the coordinates of ฮAโBโCโ after each
transformation. a. Translate ฮABC using the vector โจโ3, 1โฉ. b. Reflect ฮABC over the x-axis.
R
S
T Sโ
Tโ
Rโ
c. Reflect ฮABC over the line y = - x. d. Reflect ฮABC over the line y = -1.
17. The vertices of ฮABC are A (-3, 1), B (1, 1) and C (1, -2). Reflect ฮABC over the line x = 2. Then reflect ฮAโBโCโ over
the line y = -3. Graph ฮABC, ฮAโBโCโ, and ฮAโBโCโ. State the coordinates of ฮAโBโCโ.
Aโ:________ Bโ:________ Cโ:________
18. The coordinates of โABC are A (0, 4), B (3, 6), and C (5, 2). Graph โABC. Rotate โABC 90ยฐ, 180ยฐ, and 270ยฐ counterclockwise about the origin. Record the coordinates after each rotation.
After a 90ยฐ Rotation: Aโ ________ Bโ _______ Cโ ________
After a 180ยฐ Rotation: Aโ ________ Bโ _______ Cโ ________
After a 270ยฐ Rotation: Aโ ________ Bโ _______ Cโ ________
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 Aโ is reflected in the x-axis, the coordinates of point Aโโ are _________.
b. If point B (-4, -2) is reflected over the line y =- x, and then point Bโ is rotated 90ยฐ counterclockwise about the origin, the coordinates of Bโโ are _________.
c. If point C (6, -3) is reflected over the line y = x, and then point Cโ is rotated 270ยฐ counterclockwise about the origin, the coordinates of Cโโ are _________.
d. If point D (-2, 10) is rotated 180ยฐ about the origin, and then point Dโ is reflected over the line y=-5, then the coordinates of Dโโ are ________.
20. Point P (-6, 2) is transformed to point Pโ (2, 6). What is the transformation that maps P into Pโ? Explain. 21. The vertices of โABC are A (2, 4), B (7, 6) and C (5, 2). Graph the image of โABC after a composition of the
transformations in the order they are listed.
Transformation: (๐ฅ, ๐ฆ) โ (๐ฅ + 2, ๐ฆ โ 4) Rotation: 180ยฐ about the origin Record the coordinates of โAโBโCโ: Aโ _________ Bโ __________ Cโ __________
22. The coordinates of โPQR are: P (6, 2), Q (-6, 4), R (0, -4).
a. Draw the dilation of โPQR centered at the origin with a scale factor of 2
Pโ ________ Qโ _________ Rโ __________
b. Draw the dilation of โPQR centered at the origin with a scale factor of ยฝ
Pโ ________ Qโ _________ Rโ __________
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
125ยฐ
x 35ยฐ 85ยฐ
y
33ยฐ xยฐ
yยฐ 25ยฐ
23ยฐ
VI. Parallel Lines Intersected by a Transversal 24. Name the following pairs of angles in the diagram below:
Corresponding Angles (4 pairs) โ
Alternate Interior Angles (2 pairs) โ
Alternate Exterior Angles (2 pairs) โ
Consecutive (Same โ Side) Interior Angles (2 pairs) -
Vertical Angles (4 pairs) โ
Linear Pairs (name 4 of the 8 in the diagram) โ
25. In the figure above, if line m is parallel to line n, and the measure of angle 1 is 106ยฐ, 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
5xยฐ
7x+42ยฐ 18xยฐ
y
115ยฐ 135ยฐ
x
y
30. 31.
32. 33.
34. Angle DBA is an exterior angle of โABC. Find the measure of angle ABC.
m < ABC = ________
35. In ฮDEF, m < D = 7x + 10ยฐ, m < E = 9x -1ยฐ, and m < F = 3x + 38ยฐ. Find the measures of the angles of the triangle. What type of triangle is โDEF?
m < D = _______
m < E = _______
m < F = _______
Triangle Type: _________________
A
B C
P
Q R
90ยฐ
23ยฐ
12
5
VIII. Congruent Triangles
36. โ๐ด๐ต๐ถ โ โ๐๐๐ 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 โ Column Proofs
47. Given: ๐ด๐ตฬ ฬ ฬ ฬ โ ๐ด๐ทฬ ฬ ฬ ฬ
๐ถ ๐๐ ๐กโ๐ ๐๐๐๐๐๐๐๐ก ๐๐ ๐ต๐ทฬ ฬ ฬ ฬ
Prove: โ๐ด๐ต๐ถ โ โ๐ด๐ท๐ถ
48. Given: ๐๐ฬ ฬ ฬ ฬ ฬ ๐๐๐ ๐๐ฬ ฬ ฬ ฬ ๐๐๐ ๐๐๐ก ๐๐๐โ ๐๐กโ๐๐
Prove: โ๐๐๐ โ โ๐๐๐
49. Given: ๐ท๐ตฬ ฬ ฬ ฬ โฅ ๐ด๐ถฬ ฬ ฬ ฬ
๐ด๐ตฬ ฬ ฬ ฬ โ ๐ต๐ถฬ ฬ ฬ ฬ
Prove: < ๐ด โ < ๐ถ
Statements Reasons
N
T
M
Q
x
50. Given: ๐๐ฬ ฬ ฬ ฬ โ ๐๐ฬ ฬ ฬ ฬ ฬ ๐๐ฬ ฬ ฬ ฬ || ๐๐ฬ ฬ ฬ ฬ ฬ Prove: < ๐ โ < ๐
X. Isosceles and Equilateral Triangles
51. 52. x = ______
y = ______
๐ < 1 = ____ ๐ < 2 = ______
๐ < 3 = ______ ๐ < 4 = ________
53. 54.
a= _______ x = ________ b = ________
c = _______ d = ________
130ยฐ
y x
A
B C
S
Q R
R Q
S
55. In ฮABC, m < A = 4x + 22ยฐ, m < B = 12x - 2ยฐ, and m < C = 8x + 16ยฐ. Is ฮABC isosceles? Explain why or why not.
56. ฮQRS is isosceles with base ๐๐ ฬ ฬ ฬ ฬ . If SQ = 14x โ 3 cm, SR = 8x + 9 cm, and QR = 5x + 13 cm, find the perimeter of the triangle.
Perimeter = ____________
57. ฮQRS is isosceles with base ๐๐ ฬ ฬ ฬ ฬ . If m < Q = (6x + 3)ยฐ and m < R = (3x +21)ยฐ, 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 2880ยฐ. 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 30ยฐ. Name the polygon.
62. Each interior angle of a regular polygon has a measure of 135ยฐ. Name the polygon.