Introduction - Geometry

Transcription

1 L I F O R N I S T N R S T E S T Introduction - The following released test questions are taken from the Standards Test. This test is one of the alifornia Standards Tests administered as part of the Standardized Testing and Reporting (STR) Program under policies set by the State oard of Education. ll questions on the alifornia Standards Tests are evaluated by committees of content experts, including teachers and administrators, to ensure their appropriateness for measuring the alifornia academic content standards in. In addition to content, all items are reviewed and approved to ensure their adherence to the principles of fairness and to ensure no bias exists with respect to characteristics such as gender, ethnicity, and language. This document contains released test questions from the alifornia Standards Test forms in 2003, 2004, 2005, and First on the pages that follow are lists of the standards assessed on the Test. Next are released test questions. Following the questions is a table that gives the correct answer for each question, the content standard that each question is measuring, and the year each question last appeared on the test. The following table lists each reporting cluster, the number of items that appear on the exam, and the number of released test questions that appear in this document. Some of the released test questions for are the same test questions found in different combinations on the Integrated Mathematics 1, 2, and 3 alifornia Standards Tests and the Summative High School Mathematics alifornia Standards Test. NUMER OF NUMER OF REPORTING QUESTIONS ON RELESE TEST LUSTER EXM QUESTIONS Logic and Geometric Proofs Volume and rea Formulas ngle Relationships, onstructions, and Lines Trigonometry TOTL In selecting test questions for release, three criteria are used: (1) the questions adequately cover a selection of the academic content standards assessed on the Test; (2) the questions demonstrate a range of difficulty; and (3) the questions present a variety of ways standards can be assessed. These released test questions do not reflect all of the ways the standards may be assessed. Released test questions will not appear on future tests. For more information about the alifornia Standards Tests, visit the alifornia epartment of Education s Web site at This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

2 L I F O R N I S T N R S T E S T THE LOGI N GEOMETRI PROOFS REPORTING LUSTER The following seven alifornia content standards are included in the Logic and Geometric Proofs reporting cluster and are represented in this booklet by 22 test questions. These questions represent only some ways in which these standards may be assessed on the alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER GE1.0* GE2.0* GE3.0* GE4.0* GE5.0 GE6.0 GE7.0* Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. Students write geometric proofs, including proofs by contradiction. Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Students prove basic theorems involving congruence and similarity. Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Students know and are able to use the triangle inequality theorem. Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

3 L I F O R N I S T N R S T E S T THE VOLUME N RE FORMULS REPORTING LUSTER The following four alifornia content standards are included in the Volume and rea Formulas reporting cluster and are represented in this booklet by 11 test questions. These questions represent only some ways in which these standards may be assessed on the alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER GE8.0* GE9.0 GE10.0* GE11.0 Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders. Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

4 L I F O R N I S T N R S T E S T THE NGLE RELTIONSHIPS, ONSTRUTIONS, N LINES REPORTING LUSTER The following six alifornia content standards are included in the ngle Relationships, onstructions, and Lines reporting cluster and are represented in this booklet by 16 test questions. These questions represent only some ways in which these standards may be assessed on the alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER GE12.0* GE13.0 GE14.0* GE15.0 GE16.0* GE17.0* Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. Students prove the Pythagorean theorem. Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

5 L I F O R N I S T N R S T E S T THE TRIGONOMETRY REPORTING LUSTER The following five alifornia content standards are included in the Trigonometry reporting cluster and are represented in this booklet by 15 test questions. These questions represent only some ways in which these standards may be assessed on the alifornia Mathematics Standards Test. LIFORNI ONTENT STNRS IN THIS REPORTING LUSTER GE18.0* GE19.0* GE20.0 GE21.0* GE22.0* Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin (x)) 2 + (cos (x)) 2 = 1. Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Students know and are able to use angle and side relationships in problems with special right triangles, such as 30, 60, and 90 triangles and 45, 45, and 90 triangles. Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. * enotes key standards This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

6 L I F O R N I S T N R S T E S T 1 Which of the following best describes deductive h reasoning? using logic to draw conclusions based on accepted statements accepting the meaning of a term without definition defining mathematical terms to correspond with physical objects inferring a general truth by examining a number of specific examples h2 In the diagram below, Ð 4. SG Theorem: triangle has at most one obtuse h angle. Eduardo is proving the theorem above by contradiction. He began by assuming that in, Ð and Ð are both obtuse. Which theorem will Eduardo use to reach a contradiction? If two angles of a triangle are equal, the sides opposite the angles are equal. If two supplementary angles are equal, the angles each measure 90. The largest angle in a triangle is opposite the longest side. The sum of the measures of the angles of a triangle is 180. SG00025 l 1 2 m 3 4 t Which of the following conclusions does not have to be true? Ð 3 and Ð4 are supplementary angles. Line l is parallel to line m. Ð Ð 3 Ð Ð 3 SG10066 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

7 L I F O R N I S T N R S T E S T 4 Use the proof to answer the question below. h is the midpoint of Statement Reason is the midpoint of 1. Given 2. efinition of Midpoint 3. Reflexive Property 4.? What reason can be used to prove that the triangles are congruent? S S SS SSS SG10068 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

8 L I F O R N I S T N R S T E S T h5 In the figure below, >. 7 Which figure can serve as a counterexample to h the conjecture below? If one pair of opposite sides of a quadrilateral is parallel, then the quadrilateral is a parallelogram. rectangle rhombus square trapezoid SG10194 If we assume that mð = mð, it follows that =. This contradicts the given statement that >. What conclusion can be drawn from this contradiction? m Ð = m Ð m Ð ¹ m Ð m Ð = m Ð m Ð ¹ m Ð h 8 Given: TRP is an isosceles trapezoid with diagonals RP and T. Which of the following must be true? RP ^ T RP T T RP bisects T SG00260 SG Two lines in a plane always intersect in exactly h one point. Which of the following best describes a counterexample to the assertion above? coplanar lines parallel lines perpendicular lines intersecting lines SG00320 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

9 L I F O R N I S T N R S T E S T h 9 conditional statement is shown below. If a quadrilateral has perpendicular diagonals, then it is a rhombus. Which of the following is a counterexample to the statement above? h 10 Which triangles must be similar? two obtuse triangles two scalene triangles with congruent bases two right triangles two isosceles triangles with congruent vertex angles SG Which of the following facts would be sufficient h to prove that triangles and E are similar? SG20216 E E and E are congruent. ÐE is a right angle. and E are parallel. Ð and Ð are congruent. SG00544 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

10 L I F O R N I S T N R S T E S T h12 Parallelogram is shown below. E 14 In parallelogram FGHI, diagonals IG and FH h are drawn and intersect at point M. Which of the following statements must be true? FGI must be an obtuse triangle. HIG must be an acute triangle. FMG must be congruent to HMG. GMH must be congruent to IMF. SG00559 Which pair of triangles can be established to be congruent to prove that h15 In the figure F and and E and E and E and E SG10146 F 13 If and XYZ are two triangles such h that =, which of the following would XY YZ be sufficient to prove the triangles are similar? Ð X Ð Y Ð Z Ð Ð Y SG10218 Which additional information would be enough to prove @ E E SG This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

11 L I F O R N I S T N R S T E S T h 16 Given: and intersect at point E; Ð 2 18 In the accompanying diagram, parallel lines l h and m are cut by transversal t. t 1 l 1 3 E m Which theorem or postulate can be used to prove E E? SSS S SS SG10074 Which statement about angles 1 and 2 must be true? Ð Ð 2. Ð1 is the complement of Ð2. Ð1 is the supplement of Ð2. Ð 1 and Ð2 are right angles. SG What values of a and b make quadrilateral h MNOP a parallelogram? 17 In the figure below, n is a whole number. What h is the smallest possible value for n? N 21 O n n 3a 2b 13 M 4a + b P 15 a =1, b = SG00295 a = 5, b = a =, b = a =, b = 7 7 SG10163 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

12 L I F O R N I S T N R S T E S T h20 Quadrilateral is a parallelogram. If adjacent angles are congruent, which statement must be true? Quadrilateral is a square. Quadrilateral is a rhombus. Quadrilateral is a rectangle. Quadrilateral is an isosceles trapezoid. 22 If is a parallelogram, what is the length h of segment? E SG For the quadrilateral shown below, what is h mða + mðc? 95 o c o SG20236 a o 32 o 23 right circular cone has radius 5 inches and h height 8 inches in. l SG in. What is the lateral area of the cone? (Lateral area of cone = prl, where l = slant height) 40p sq in. 445p sq in. 5p 39 sq in. 5p 89 sq in. SG00053 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

13 L I F O R N I S T N R S T E S T h24 Figure is a kite. 26 The four sides of this figure will be folded up h and taped to make an open box. 5 centimeters 10 cm 8 cm 17 cm 6 cm 15 cm 10 cm 8 cm 17 cm What is the area of figure, in square centimeters? SG If a cylindrical barrel measures 22 inches in h diameter, how many inches will it roll in 8 revolutions along a smooth surface? What will be the volume of the box? 50 cm 3 75 cm cm cm 3 SG p in. 168p in. 176p in. 228p in. SG00564 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

14 L I F O R N I S T N R S T E S T h27 classroom globe has a diameter of 18 inches. 28 The rectangle shown below has length h 20 meters and width 10 meters. 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m Which of the following is the approximate surface area, in square inches, of the globe? (Surface rea = 4pr 2 ) SG20238 If four triangles are removed from the rectangle as shown, what will be the area of the remaining figure? 136 m m m m 2 29 If RSTW is a rhombus, what is the area of h WXT? SG00012 R W X o S T SG00227 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

15 L I F O R N I S T N R S T E S T 30 What is the area, in square units, of the h trapezoid shown below? y (0, 5) (8, 5) 0 (12, 0) The figure below is a square with four h congruent parallelograms inside. x SG The perimeters of two squares are in a ratio of h 4 to 9. What is the ratio between the areas of the two squares? 2 to 3 4 to 9 16 to to 81 SG00013 h33 Lea made two candles in the shape of right rectangular prisms. The first candle is 15 cm high, 8 cm long, and 8 cm wide. The second candle is 5 cm higher but has the same length and width. How much additional wax was needed to make the taller candle? 320 cm cm cm cm SG h34 Two angles of a triangle have measures of 55 and 65. Which of the following could not be a measure of an exterior angle of the triangle? What is the area, in square units, of the shaded portion? SG00571 SG20225 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

16 L I F O R N I S T N R S T E S T h35 The sum of the interior angles of a polygon is the same as the sum of its exterior angles. What type of polygon is it? quadrilateral hexagon octagon decagon h36 What is mð x? x SG00305 h38 In the figure below,. (x + 40)º (x 40)º What is the value of x? SG h39 What is mð 1? 36 o 88 o SG If the measure of an exterior angle of a regular h polygon is 120, how many sides does the polygon have? SG SG20179 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

17 L I F O R N I S T N R S T E S T h40 What is mð WZX? 41 diagram from a proof of the Pythagorean h theorem is pictured below. 132º V a b W 52º Z c Y 80 X SG30022 Which statement would not be used in the proof of the Pythagorean theorem? 1 The area of a triangle equals ab. 2 The four right triangles are congruent. The area of the inner square is equal to half of the area of the larger square. The area of the larger square is equal to the sum of the areas of the smaller square and the four congruent triangles. SG10192 h42 right triangle s hypotenuse has length 5. If one leg has length 2, what is the length of the other leg? SG00566 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

18 L I F O R N I S T N R S T E S T 43 new pipeline is being constructed to re-route h its oil flow around the exterior of a national wildlife preserve. The plan showing the old pipeline and the new route is shown below. 44 Marsha is using a straightedge and compass to h do the construction shown below. 32 miles 60 miles NEW PIPELINE OL PIPELINE P bout how many extra miles will the oil flow once the new route is established? l SG10016 Which best describes the construction Marsha is doing? a line through P parallel to line l a line through P intersecting line l a line through P congruent to line l a line through P perpendicular to line l SG00526 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

19 L I F O R N I S T N R S T E S T h45 Given: angle What is the first step in constructing the angle bisector of angle? 46 Scott is constructing a line perpendicular to h line l from point P. Which of the following should be his first step? P l P raw ray. l raw a line segment connecting points and. From points and, draw equal arcs that intersect at. From point, draw an arc that intersects the sides of the angle at points and. P l SG10131 P l SG00308 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

20 L I F O R N I S T N R S T E S T h47 The diagram shows. y h48 Figure O is a parallelogram. y (a, b) (a + c, b) x O (c, 0) x Which statement would prove that is a right triangle? What are the coordinates of the point of intersection of the diagonals? ( slope )( slope ) = 1 ( slope )( slope ) = -1 distance from to = distance from to distance from to = - (distance from to ) SG00475 æ a b ö, ç è 2 2 ø æ c b ö ç è2, 2 ø ç çæ a + c, b ö çè 2 2 ø æ a + c a + bö, ç è 2 2 ø SG This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

21 L I F O R N I S T N R S T E S T h 49 What type of triangle is formed by the points (4, 2), (6,-1), and (-1, 3)? right equilateral isosceles scalene h 51 In the figure below, sin = SG In the figure below, if sin x = 5, what are h cos x and tan x? 13 x cos = 12 5 x and tan x = cos x = 12 and tan x = cos = 13 5 x and tan x = cos x = 13 and tan x = SG00493 What is the length of? SG00432 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

22 L I F O R N I S T N R S T E S T 52 pproximately how many feet tall is the h streetlight? h53 In the accompanying diagram, mð = 32 and = 10. Which equation could be used to find x in? h x 20 ft 40º sin 40» 0.64 cos 40» 0.77 tan 40» x =10 sin 32 x =10 cos x =10 tan x = cos32 SG20047 SG00555 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

23 L I F O R N I S T N R S T E S T h54 The diagram shows an 8-foot ladder leaning against a wall. The ladder makes a 53 angle with the wall. Which is closest to the distance up the wall the ladder reaches? h55 Triangle JKL is shown below. K 24 o L ft 4.8 ft 8 ft sin cos tan ? J Which equation should be used to find the length of JK? JK sin 24 = 28 sin 24 = 28 JK cos 24 = JK 28 cos 24 = 28 JK SG ft 9.6 ft 56 If a = 3 3 in the right triangle below, what is h the value of b? SG00342 a b c SG10052 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

24 L I F O R N I S T N R S T E S T h57 What is the value of x in the triangle below? 59 In the circle below, and are chords h intersecting at E. 10 x x E SG square is circumscribed about a circle. What h is the ratio of the area of the circle to the area of the square? If E = 5, E = 12, and E = 6, what is the length of E? p p 4 SG00022 SG00585 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

25 L I F O R N I S T N R S T E S T h60 R is tangent to a circle, whose center is, at point. is a diameter. R h 61 In the figure below, is tangent to circle O at point, secant intersects circle O at points and, m = 70, and m = o 25º O 110 o What is mð R? SG20186 What is mð? SG10257 h62 The vertices of are (2, 1), (3, 4), and (1, 3). If is translated 1 unit down and 3 units to the left to create EF, what are the coordinates of the vertices of EF? (0, 1), E(1, 2), F(1, 3) (0, 1), E(0, 3), F( 2, 2) ( 2, 2), E(0, 3), F( 1, 0) ( 1, 0), E(0, 3), F( 2, 2) SG00317 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

26 L I F O R N I S T N R S T E S T h63 If triangle is rotated 180 degrees about the origin, what are the coordinates of? ( ) ( 5, 4 ) ( 4, 5 ) ( 4, 5) -5, y x SG10096 h 64 Trapezoid below is to be translated to trapezoid by the following motion rule. ( x, y ) x + 3, y ( ) y What will be the coordinates of vertex? ( 1, -3) ( 2, 1 ) ( 6, 1 ) ( 8, -3) x SG10214 This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

27 L I F O R N I S T N R S T E S T Question Number orrect nswer Standard Year of Test GE1.0 GE1.0 GE2.0 GE2.0 GE2.0 GE3.0 GE3.0 GE3.0 GE3.0 GE4.0 GE4.0 GE4.0 GE4.0 GE4.0 GE5.0 GE5.0 GE6.0 GE7.0 GE7.0 GE7.0 GE7.0 GE7.0 GE8.0 GE8.0 GE8.0 GE9.0 GE9.0 GE10.0 GE10.0 GE10.0 GE10.0 GE11.0 GE This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.

28 L I F O R N I S T N R S T E S T Question Number orrect nswer Standard Year of Test 34 GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE GE This is a sample of alifornia Standards Test questions. This is NOT an operational test form. Test scores cannot be projected based on performance on released test questions. opyright 2007 alifornia epartment of Education.