GEOMETRY MIDYEAR REVIEW (TOPICS AND PROBLEMS)
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1 GEOMETRY MIDYEAR REVIEW (TOPICS AND PROBLEMS) Algebra how to solve a quadratic equation how to solve a system of equations: by substitution how to simplify square roots by addition/elimination how to solve an equation with fractions by clearing the fractions Angles Complementary angles add to 90 Supplementary angles add to 180 Adjacent angles share a side (so that they re back-to-back) Linear pair of angles are adjacent, form a straight line, and are supplementary Vertical angles are congruent Corresponding angles are congruent (but only if 2 lines ) Alternate interior angles are congruent (but only if 2 lines ) Same side interior angles are supplementary (but only if 2 lines ) An angle bisector splits an angle in half Lines Equations of lines: y = mx + y-intercept y ycoord = m (x xcoord) x = xcoord for a vertical line (slope undefined) y = ycoord for a horizontal line (slope of 0) y-intercept is where line crosses y-axis (make x = 0 to find it) x-intercept is where line crosses x-axis (make y = 0 to find it) midpoint of a segment: (average the x-coords, average the y-coords) slope of a line or segment = rise run = y 2 y 1 x 2 x 1 describes how steep a line is length of a segment = rise 2 + run 2 describes how long a segment is parallel lines have same slope perpendicular lines have negative reciprocal slopes how to find the equation of the perpendicular bisector of a segment
2 Circles (x xcord center) 2 + (y ycoord center) 2 = radius 2 how to find the equation of a circle given center and radius how to find the equation of a circle given center and point on circle how to find the equation of a circle given endpoints of diameter Triangles Median of a triangle joins the midpoint of the opposite side Altitude of a triangle is perpendicular to the opposite side Total degrees of the three angles = 180 exterior angle = far interior angle + far interior angle isosceles triangles: if 2 sides congruent, then base angles congruent if 2 angles congruent, then legs are congruent ways to show that two triangles are congruent: SSS, SAS, ASA, AAS, HL Polygons sum of interior angles = 180 (n 2) sum of exterior angles = 360 if the polygon is regular, then all angles are the same, so: each interior angle = each exterior angle = sum of interior angles n sum of exterior angles n
3 Part I - Multiple Choice. Circle your answer. 1. Solve 2 3 x 2 = 3 5 x A. 30 B. 2 C. 30/19 D Which picture is a ray? A. B. C. D. 3. AM = 3 x, BM = 2x and AB = 22. Find x. 4 A M B A. 2 B. 4 C. 6 D Solve: x + 5y = 8 3x 2y = 7 A. (1, 3) B. (1, 4) C. (3, 1) D. (6, 1) and 2 are: 2 A. vertical B. supplementary C. complementary D. adjacent 6. The midpoint of the segment joining the points (3, 5) and ( 1,0) is: A. (2, 5) B. (1, 2.5) C. (2, 2.5) D. (2, 3) = A. 7 B. 14 C. 7 D. 49
4 8. How many points does a line contain? A. 1 B. 2 C. 3 D. infinite 9. Which of the following points on the number line is closest to 94? W X Y Z A. W B. X C. Y D. Z 10. A C T Which one of these is a name for the angle shown? A. CAT B. CTA C. TAC D. ACT 11. The slope of the line 3x + y = 6 is: A. 3 B. 3 C. 1/3 D The slope of the line 7y 9x = 3 is: A. 7 B. 7/9 C. 9/7 D. 9/7 13. Find an equation of a vertical line passing through the point (4, 3). A. x = 4 B. x = 3 C. y = 4 D. y = 3
5 Use the diagram to the right for #14 16: 14. In the following diagram, AL bisects KAT. If LAT = 20, then 3 is: A. 20 B. 40 C. 50 D If TAK = 80, then 2 is: A. 20 B. 40 C. 50 D If 3 = 130, find 1 is: A. 25 B. 50 C. 65 D M is the midpoint of AB. AM = 2 x and MB = 24. Find x. 3 A M B A. 16 B. 18 C. 12 D Find the measure of b: A. 106 B. 74 C. 37 D
6 20. Which angle forms a linear pair with TWV? A. RWS B. TWU C. VWR D. VWU 21. To graph the line y = 3x 4, you need to start by graphing which point? A. (0, 4) B. (0, 4) C. (0, 3) D. (0, 3) From there, you should go where, to graph a second point? A. up 3 and left 1 B. up 32 and right 1 C. up 1 and left 3 D. up 1 and right What is the slope of a line whose x intercept is 2 and whose y intercept is 1? A. 2 B. 2 C. 1/2 D. 1/2 23. To graph the line y 2 = 1 ( 2 x +1 ), you need to start by graphing which point? A. (1, 2) B. (1, 2) C. ( 1, 2) D. ( 1, 2) 24. Find the distance between the points (3, 2) and (3, 6). A. 8 B. 8 C. 17 D. 65
7 Part II Short answer. Show your work. 1. Given two parallel lines cut by a transversal, find x and y. 2x y x ) x = y = 3) In the diagram, a b a) 1 = 10x 30 7 = 5x + 40 Relationship: x = y = x = b) 4 = x 2 + 6x 5 = 3x + 18 Relationship: x = or x =
8 4) Lila is out on the sand dunes hang gliding again. On her last flight, she jumps off a cliff and travels in a diagonal line for 500 feet before landing 450 feet from the base of the cliff. (a) (b) Put the numbers into the diagram in the appropriate places. How high is the cliff? 6) Write in the measures of all the unknown angles. b d a 54 c e a = b = c = d = e = 7) Find the equation of a line passing through (2, 1) and parallel to the line 3x + y = 6. 8) Find the equation of a line passing through (2, 1) and perpendicular to the line 7y 9x = 3 9) Find an equation of the line passing through the given points ( 5, 6) and ( 4, 3).
9 10) y = 1 4 x 1 Point on the line: slope: 11) Graph the line: y 2 = 1 2 ( x +1) Point on the line: slope: 12) The measure of an angle is 5 times the measure of its complement. Find the measure of the angle and its complement. 13) The measure of an angle is 5 more than 4 times the measure of its supplement. Find the measure of the angle and its supplement. 14) Simplify each: ( 5 ) ( ) ( ) ( 5 4 3)
10 15) G = 50 (a) (b) (c) (d) (e) If H form a linear pair with G, what is the measure of H? If H and G are complementary, what is the measure of H? If H and G are supplementary, what is the measure of H? If H and G are vertical angles, what is the measure of H? If H and G are adjacent angles, what is the measure of H? now assume that there are 2 parallel lines to answer the following: (e) (f) (g) If H and G are corresponding angles, what is H? If H and G are alternate interior angles, what is H? If H and G are same-side interior angles, what is H? 16) 1 2 If 1 = 2x + 3 and 2 = x 6, then: x = 1 = 2 = 17) B D CD bisects BCE BCD = x 2 +2x DCE = 35 C E x = or
11 18) Can 6, 8 and 14 be the sides of a right triangle? SHOW WORK!! Part III Short answer. Show your work. 19) x = or 20) Given the two points: (3, 2) and (4, 8), find the midpoint, slope and distance. Picture with points labeled: midpoint = slope = distance (or length) = equation for this line: 21) Find x. x 5 x = x =
12 22) 23) 24) Find the measure of each exterior angle for a regular 23-gon.
13 25) 26) Find the measure of each interior angle for a regular 21-gon.
14 27) 31) <1 = 32) How many sides does a polygon have if the sum of its interior angles is 2880? 33) Find the number of sides of a regular polygon if each of its exterior angles is 2. 34) Find the sum of the measures of the interior angles of a nonagon. 35) Find the sum of the measures of the exterior angles of a 2006-gon.
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