INTEGRATED MATH 2: TRIMESTER A FINAL REVIEW #1

Transcription

1 INTEGRATED MATH 2: TRIMESTER A FINAL REVIEW #1 1) Based on the markings, name each polygon below with the most specific name. Note: The polygons are not drawn to scale. c) 2) Place the polygons from your Polygon Graphic Organizer into the appropriate regions of the Venn diagram at right. The conditions that the polygons must meet to be placed in each circle are listed in each problem. Note: Create a new Venn diagram for each problem. a) Circle #1: Has more than three sides Circle #2: Has at least one pair of parallel sides b) Circle #1: Has fewer than four sides Circle #2: Has at least two sides equal in length c) Circle #1: Is equilateral Circle #2: Is equiangular 3) Write an equation showing area as a product equals area as a sum. c) 4) Multiply. (Hint: you may want to use an area model) a) (3x + 2)(2x + 7) b) (2x)(x 1) c) (x 1)(x + y + 1) 5) Can the triangle have side lengths of: a) 1, 2, 3? b) 7, 8, 9? c) 4.5, 2.5, 6? d) 9.5, 1.25, 11.75? 6) Use what you know about angle measures to determine the values of x and y.

2 7) Use the geometric properties and theorems you have learned to solve for x in each diagram. c) d) e) f) g) h) 8) Decide if each of the following pairs of triangles must be congruent. If so, state the triangle congruence theorem that supports your conclusion. c) d) e) f) 9) For the pair of triangles below, decide whether they are congruent, not congruent, or there is not enough information to determine their congruence. Justify your argument with a flowchart

3 10) Write the converse of the statement and state whether or not the converse is true. a) If one angle of a quadrilateral is a right angle, then the quadrilateral is a rectangle. b) If two angles of a triangle have equal measures, then the two sides of the triangle opposite those angles have equal length. c) If an angle is a straight angle, then the angle measures ) Use the method of proof by contradiction to justify each of your conclusions to problems. a) Nik scored 40 points lower than Tess on their last math test. The scores could range from 0 to 100 points. Could Tess have scored a 30 on this test? Justify using a proof by contradiction. b) Jolie claims that a triangle can have two right angles. Prove her wrong! Justify your answer with a proof by contradiction. 12) For each pair of similar figures below, determine the ratio of similarity for large: small. Diagrams are drawn roughly to scale; you can assume that sides that look longer are longer. 13) For each pair of similar figures, state the ratio of similarity. Then use it to calculate the value of x. Diagrams are drawn roughly to scale; you can assume that sides that look longer are longer. 14) Each pair of figures below is similar. Use what you know about similarity to solve for x. Diagrams are drawn roughly to scale; you can assume that sides that look longer are longer.

4 15) Solve for the missing lengths in the pairs of similar figures below. 16) Decide if each pair of triangles is similar. If they are similar, write a correct similarity statement and justify your answer. c) d) 17) Using the information given in each diagram below, decide if any triangles are congruent, similar but not congruent, or not similar. If you claim the triangles are congruent or similar, create a flowchart justifying your answer.

5 18) Standing 4 feet from a mirror resting on the flat ground, Palmer, whose eye height is 5 feet, 9 inches, can see the reflection of the top of a tree. He measures the mirror to be 24 feet from the tree. How tall is the tree? Draw a picture to help solve the problem. 19) For each problem, write an equation to represent the situation and then solve. Be sure to define your variable(s) and clearly answer the question. a) Nancy started the year with $435 in the bank and is saving $25 a week. Shane started with $875 and is spending $15 a week. When will they both have the same amount of money in the bank? b) Leticia spent $11.19 on a bag of red and blue candy. The bag of candy weighed 11 pounds. If the red candy costs $1.29 per pound and the blue candy costs $0.79 per pound, how much of each candy did she buy? c) Determine three consecutive numbers whose sum is ) A concert has been sold out for weeks, and as the date of the concert draws closer, the price of the ticket increases. The cost of a pair of tickets was $150 yesterday and is $162 today. Assume that the cost continues to increase at this rate exponentially. a) What is the daily rate of increase? What is the multiplier? b) What will be the ticket cost one week from now, the day before the concert? c) What was the cost two weeks ago? 21) Ryan s motorcycle is now worth $2500. It has decreased in value 12% each year since it was purchased. If he bought it four years ago, what did it cost new? 22) Determine the area and perimeter of each figure. Note: All angles that look like right angles can be assumed to be right angles. 23) Plot triangle ABC with vertices A(0, 0), B(3, 4), and C(3, 0) on graph paper. Using the origin as the point of dilation, enlarge it by a factor of 2.

6 SOLUTIONS 1) a) Parallelogram b) Isosceles Triangle c) Isosceles Trapezoid 2) a) Common to both circles and placed in the overlapping region are: square, rectangle, parallelogram, isosceles trapezoid, trapezoid, right trapezoid, rhombus, and regular hexagon Only in Circle #1: quadrilateral, kite, and regular pentagon Only in Circle #2: none Outside of both circles: scalene triangle, equilateral triangle, isosceles right triangle, isosceles triangle, and scalene right triangle b) Common to both circles and placed in the overlapping region are: equilateral triangle, isosceles triangle, and isosceles right triangle Only in Circle #1: scalene triangle and scalene right triangle Only in Circle #2: square, rectangle, parallelogram, rhombus, kite, regular pentagon, isosceles trapezoid, and regular hexagon Outside of both circles: quadrilateral, trapezoid, and right trapezoid c) Common to both circles and placed in the overlapping region are: equilateral triangle, square, regular pentagon, and regular hexagon. Only in Circle #1: rhombus Only in Circle #2: rectangle Outside of both circles: isosceles right triangle, isosceles triangle, scalene right triangle, scalene triangle, parallelogram, quadrilateral, kite, isosceles trapezoid, right trapezoid, trapezoid 3) a) (x 5)(x + 3) = x 2 2x 15 b) 6(3y 2x) = 18y 12x c) (x + 4)(3y 2) = 3xy 2x + 12y 8 4) a) 6x x + 14 b) 2x 2 2x c) x 2 + xy y 1 5) a) no b) yes c) yes d) no 6) a) x = 19, y = 110 b) x = 25, y = 90 7) a) 45 b) 15 c) 20 d) 7 e) 9 f) 2 g) 65 h) 43/6 8) a) ABC DEF by ASA b) PNM PNO by SSS c) The triangles are not necessarily congruent. d) GI GI, so GHI IJG by SSS e) Vertical angles, so POQ ROS by SAS f) No, the lengths of the hypotenuses of the triangles are different.

7 9) 10) a) Converse: If a quadrilateral is a rectangle, then one angle is a right angle. True, in fact, all four angles are right angles. b) Converse: If two sides of a triangle are equal in length, then the two angles opposite those sides are equal in measure. True. c) Converse: If an angle measures 180, then it is a straight angle. True. 11) a) Assume that Tess scored 30 points. Then Nik s score was = 10, which is impossible. So Tess cannot have a score of 30 points. 7. b) Assume that a triangle has two right angles. Using the Triangle Sum Theorem, the measure of the third angle must be zero. However, this is impossible, so a triangle cannot have two right angles. OR: If a triangle has two 90 angles, the two sides that intersect with the side between them would be parallel and never meet to complete the triangle, as shown in the figure. 12) a) 5 b) 15/8 = ) a) 2; x = 72 b) 3/2; x = ) a) x = 9 b) x = 40/ ) a) x = 1.25 b) x = 12 16) a) BOX ~ NCA by AA ~ b) The triangles are not similar because the sides are not proportional. 12/15 = 18/22.5 = 0.8, but 10/ c) ALI ~ MES by SAS ~ d) The triangles are not similar. On SAM, the 60 is included between the two given sides, but on UEL the angle is not included. 17) a) b)

8 18) 19) a) Let x represent the number of weeks, $435 + $25x = $875 $15x; 11 weeks b) Let R represent pounds of red candy and B represent pounds of blue candy, B + R = 11 and $1.29R + $0.79B = $11.19; 5 pound of red, 6 pounds of blue c) Let x represent the smallest number, x + (x + 1) + (x + 2) = 219; 72, 73, 74 20) a) 8%, 1.08 b) $ c) $ ) $2500 = a(0.88) 4 ; a $ ) a) A = 294 ft 2, P = 74 ft b) A = 49 in 2, P = 47.2 in 23)