2013 2014 Geometry Semester 2 Instructional Materials for the WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Math Common Final blueprint for this course. When used as test practice, success on the Instructional Materials does not guarantee success on the district math common final. Students can use these Instructional Materials to become familiar with the format and language used on the district common finals. Familiarity with standards vocabulary and interaction with the types of problems included in the Instructional Materials can result in less anxiety on the part of the students. Teachers can use the Instructional Materials in conjunction with the course guides to ensure that instruction and content is aligned with what will be assessed. The Instructional Materials are not representative of the depth or full range of learning that should occur in the classroom Note from Earl. Many of the solutions in this document use techniques presented in the Geometry Handbook, which is available on the www.mathguy.us website. If you have trouble following any of the techniques used, try looking in the handbook for pages that deal with the issue you are struggling with. I solve the problems in this sample test using the quickest method available in most cases. Occasionally, I also make comments about some of the math involved in an effort to enhance your understanding of what is going on in the problem. You may have learned different techniques in your classroom. Use whichever techniques work for you. Finally, if there is a conflict between the content of this document and what you have learned in class, your teacher should be the primary source for how any problem should be solved. 1 Page
Multiple Choice: Identify the choice that best completes the statement or answers the question. Figures are not necessarily drawn to scale. 1. The ratios of the areas of square A to square B is. If the area of square B is 100, what is the length of a side of square A? A. 4 C. 10 B. 8 D. 64 For this kind of problem, we set up a proportion, with the ratio of the areas on one side and the actual areas on the other. 16 25 100 Cross multiply the two fractions to get: 16 100 25. Then: 64. We are not yet finished because the problem asks for the length of the side of Square. For this, we use the area formula, where represents the length of a side: 64, which we solve to obtain. Answer B 2. What is the scale factor for the dilation of to image? A. 2 B. 1 C. 2 D. 3 To determine the scale factor of a dilation from the origin, pick corresponding points on the two figures and divide the and values of the image by those of the pre image. If the two values that result are the same, you have most likely determined the scale factor without error. I have selected Points 1, 4 and 3, 12 for this purpose. 3 12 1 4 2 Page
Since the calculated scale factors are both, I can conclude with confidence that Answer D is correct. 3. Apply the dilation :, 4, 4 to the polygon with the given vertices. Name the coordinates of the image points. A. 8, 4, 16, 4, 16, 12 B. 8, 4, 16, 4, 16, 12 C. 0.5, 0.25, 1, 0.25, 1, 0.75 D. 0.5, 0.25, 1, 0.25, 1, 0.75 Determine the coordinates of the points; then apply the dilation, as follows: 2,1 4,1 4,3 4 4 4 2, 4 1 8, 4 4, 4 1 16,4 4, 4 3 16, 12 Answer A 4. The pair of triangles is similar. What is the value for? A. 18 B. 16 C. 13 D. 3 To help with this problem, let s assign some letters to the vertices of the two triangles. Then, we note that ~ by following the angle congruences from angle to angle ( ; ; ). is the measure of side, so we want it in our proportion. Then, using corresponding sides, we have: 9 6 12 108 6 Answer A The following page contains information about the similarity theorems from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. 3 Page
Geometry Similar Triangles The following theorems present conditions under which triangles are similar. Side Angle Side (SAS) Similarity Side Side Side (SSS) Similarity SAS similarity requires the proportionality of two sides and the congruence of the angle between those sides. Note that there is no such thing as SSA similarity; the congruent angle must be between the two proportional sides. SSS similarity requires the proportionality of all three sides. If all of the sides are proportional, then all of the angles must be congruent. Angle Angle (AA) Similarity AA similarity requires the congruence of two angles and the side between those angles. Similar Triangle Parts In similar triangles, Corresponding sides are proportional. Corresponding angles are congruent. Establishing the proper names for similar triangles is crucial to line up corresponding vertices. In the picture above, we can say: ~ or ~ or ~ or ~ or ~ or ~ All of these are correct because they match corresponding parts in the naming. Each of these similarities implies the following relationships between parts of the two triangles: and and 4 Page
5. In the figure, ~. What is the length of side? A. 6 B. 7 C. 16 D. 20 Note the angle congruences that result from the parallel lines: and Then, we note that ~ from the AA Similarity Theorem. Then, using corresponding sides, we have the following proportion: 16 4 4 8 5 80 16 32 112 16 7 Finally, we want to determine: AB 4 8 4 7 8. Answer D 6. Two triangles are similar and the ratio of each pair of corresponding sides is 2: 1. Which statement regarding the two triangles is not true? A. Their areas have a ratio of 4: 1 B. The scale factor is a ratio of 2: 1 C. Their perimeters have a ratio of 2: 1 D. Their corresponding angles have a ratio of 2: 1 Some thoughts: If the ratio of side lengths is : in a pair of similar figures. Then: The measures of the angles in the two figures are the same. The ratio of any one dimensional measurements is :. The ratio of any two dimensional measurements (e.g., areas) is : The ratio of any three dimensional measurements (e.g., volumes) is : Relating these rules to this problem, we note that the scale factor is one dimensional, so A, B, and C are all true. D is false because the angle measures in the triangles are the same. Answer D 5 Page
7. What is the value of? A. 39 B. 54 C. 63 D. 90 To help with this problem, let s assign some letters to the vertices of the two triangles. Then, we note that ~. is the measure of side, so we want it in our proportion. Using corresponding sides, we have: 921 9 27 30 27 9 Answer D 8. In, is the midpoint of, is the midpoint of, and R is the midpoint of. Given the following, what is the perimeter of? 8.2. 4.3 6 A. 11.4 B. 15.5 C. 18.4 D. 18.5. The Perimeter of is: following: 1 2 1 2 63 4.3 (given) 8.2. Since points, and are midpoints, we have the The perimeter then is: 3 4.3 8.2. Answer B 6 Page
9. In the figure, is a right triangle and ~. What is the length of? A. 20 B. 40 C. 2 10 D. 10 20 There are special rules for these three triangle problems, which may be learned as words instead of formulas. I list all three here because the problem on the final may involve different parts of the three triangles. The height squared = the product of: the two parts of the base The left side squared = the product of: the part of the base below it and the entire base The right side squared = the product of: the part of the base below it and the entire base Using the first column of this table, we have: or 5 840 Then, taking the square root of both sides of the equation, we get:. Answer C 10. In the right triangle, and represent unknown side lengths. What is the length of side? A. 2 B. 4 C. 2 3 D. 3 2 In a Triangle, the proportions of the sides are: : : for short side : long side : hypotenuse respectively. In this problem, we are given the short side and asked to calculate the hypotenuse. The length of the hypotenuse is two times the length of the short side. So, we have: 2 2. Answer B 7 Page
11. In the figure, what is the distance a ball travels when thrown from second base to home plate? A. 90 B. 180 C. 90 2 D. 2 90 In a Triangle, the proportions of the sides are: : : for side : side : hypotenuse respectively. In this problem, we are given the two sides and asked to calculate the hypotenuse. The length of the hypotenuse is 2 times the size of the length of a side. So, we have: 2 90 feet. Answer C 12. What is cos in the triangle? A. 40 41 C. 41 9 B. 9 41 D. 41 40 SOH CAH TOA sin cos tan In this problem: cos Answer A 8 Page
13. What is the value of in the triangle? A. 16 cos 35 cos 35 B. 16 C. 16 sin 35 D. 16 sin 35 Using SOH CAH TOA, we first note that relative to the angle of 35, we are given the opposite side and the hypotenuse. The trig function that uses these two sides is the sine function. So, sin 35 sin 35 16 x 16 sin 35 Answer D 14. What is the measure of angle A in the triangle, rounded to the nearest degree? A. 35 B. 44 C. 46 D. 72 Using SOH CAH TOA, we first note that relative to angle, we are given the adjacent side and the hypotenuse. The trig function that uses these two sides is the cosine function. So, cos cos 1 7 10 ~ (using a calculator) Answer C 9 Page
15. A person is standing at ground level with the base of the Empire State Building in New York City. The angle formed by the ground and a line segment from his position to the top of the building is 48.4. The height of the Empire State Building is 1472 feet. Find the distance that he is standing from the base of the Empire State Building to the nearest foot. A. 8 C. 1968 B. 1307 D. 2217 We need a picture for this one, so I drew the one to the right. Using SOH CAH TOA, we first note that relative to the 48.4 angle, we are given the adjacent side and the opposite side. The trig function that uses these two sides is the tangent function. So, t an 48.4,,. ~, Answer B 16. In the figure, is a right triangle with the hypotenuse. Given the segments lengths of 5, 4,2,and 1, what is cos? A. 5 C. 5 5 B. 12 5 25 D. 5 5 12 I have added the given dimensions to the diagram, as You should on any problem that gives lengths or angles. Next, notice that calculating the cosine of angle B requires us to use only at the right. Using SOH CAH TOA, cos, but we do not want to leave a radical in the denominator, so, cos 1 5 1 5 5 5 5 5 Answer C 10 Page
17. Using a string a student decided to determine the diameter of a large trash can. If a string 60 long will wrap around the trash can, what is the approximate diameter of the trash can? A. 25 C. 9.55 B. 19.1 D. 9 Drawing a picture of the situation helps see the one at right. Next, we need the formula for circumference in terms of diameter: In this problem, 60, so we get: 60 ~. inches Answer B 18. A rectangle is inscribed in a circle as shown below. Find the exact circumference of the circle. A. 13 B. 17 C. 119 D. 169 Draw the diameter in the picture to the right. Calculate using the Pythagorean Theorem: 5 12 25 144 169. Taking the square root of both sides gives: 13. Next, we need the formula for circumference in terms of diameter: In this problem, 13, so we get: 13 mm Answer A 11 Page
19. What is the area of a regular hexagon with an apothem of 10 and a side length of? A. 200 3 B. 300 3 C. 600 3 D. 600 We need the formula for area of a regular polygon based on the apothem and side length:, where: length of apothem, perimeter of the regular polygon. In this problem, 10 and 6 40 3, so we get: 1040 3 cm 2 Answer A 20. Given that the side of the regular pentagon is 8 and that the apothem is approximately 5.51, what is the approximate area of the shaded triangle? A. 20 B. 22 C. 64 D. 110 We want the area of the shaded region, which is a triangle. So, we will use the formula for area of a triangle., where: length of the base, height of the triangle. Note that the base of the triangle is the length of a side of the polygon ( 8 ft. ). Also, the height of the triangle is the length of the apothem (5.51 ft.). In this problem, 8 and 5.51, so we get: 85.51 ~ ft 2 Answer B 12 Page
21. If two pieces of ice have the same volume, the one with the greater surface area will melt faster because more of its surface area is exposed to the air, which is warmer than the ice. Two pieces of ice labeled and B have the same volume. Each piece of ice is shaped like a rectangular prism. Which piece of ice melts the fastest? A. Piece melts the fastest. B. Piece melts the fastest. C. They take the same amount of time to melt. D. A relationship cannot be determined First check that the volumes are equal: 32 3 3 288 ft 3 18 8 2288 ft 3 Since the volumes are equal, we can check the surface areas: 2 2 2 2323 2323 233 402 ft 2 2188 2182 282 392 ft 2 The one that melts fastest is the one with the greatest surface area, which is Piece. Answer A 22. What best describes the cross section shown on the cube? A. square C. trapezoid B. triangle D. rectangle The cross section shown is a quadrilateral. It appears to have two parallel sides (the top and bottom) and two non parallel sides (the left and right). This describes a trapezoid. Answer C The following page contains information about the various kinds of quadrilaterals from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. 13 Page
Geometry Figures of Quadrilaterals Kite 2 consecutive pairs of congruent sides 1 pair of congruent opposite angles Diagonals perpendicular Trapezoid 1 pair of parallel sides (called bases ) Angles on the same side of the bases are supplementary Isosceles Trapezoid 1 pair of parallel sides Congruent legs 2 pair of congruent base angles Diagonals congruent Parallelogram Both pairs of opposite sides parallel Both pairs of opposite sides congruent Both pairs of opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangle Parallelogram with all angles congruent (i.e., right angles) Diagonals congruent Rhombus Parallelogram with all sides congruent Diagonals perpendicular Each diagonal bisects a pair of opposite angles Square Both a Rhombus and a Rectangle All angles congruent (i.e., right angles) All sides congruent 14 Page
23. A layered cake is a solid of revolution. Which of the following is the drawing of a twodimensional shape and an axis of rotation that could form the cake? A. C. B. D. Rotating a straight edge creates a circular shape in three dimensions. We want half of the shape that we see looking at a cross section of the cake in 2 dimensions. Answer D 24. What is the volume of the cylinder in terms of x? A. 15 12 B. 15 12 C. 45 36 D. 75 120 489 We need the formula for volume of a cylinder based on the radius of a base and the height:, where: radius of a base, height of the cylinder. In this problem, 3 and 54, so we get: 35 4 9 5 4 cm 3 Answer C The following page contains formulas for the volumes and surface areas for various solids from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. 15 Page
Geometry Summary of Surface Area and Volume Formulas 3D Shapes Shape Figure Surface Area Volume Sphere Right Cylinder Cone Square Pyramid Rectangular Prism Cube General Right Prism 16 Page
25. What is the height of a square pyramid that has a side length of 13 and a volume of 1521? A. 3 C. 27 B. 9 D. 39 We need the formula for volume of a square pyramid based on the length of a side of the base and the height:, where: length of a side of the base, height of the pyramid. In this problem, 13 and 1,521, so we get: 1,521 13 1,521 Solving for, we get:, ft. Answer C 26. A food manufacturer sells yogurt in cone shaped cups with the dimensions shown. To the nearest tenth, how many fluid ounces of yogurt does the cup hold? (Hint: 1 0.034 ) A. 0.6 B. 5.7 C. 17.1 D. 22.8 We need the formula for volume of a cone based on the radius of the base and the height:, where: radius of a base, height of the cone. In this problem, 824 and 10, so we get: 4 10 160 ~ 167.55 cm3 In fluid ounces, this is ~ 167.550.034 ~. fl. oz. Answer B 17 Page
27. What is the volume of the sphere in terms of? A. 36 B. 48 C. 288 D. 864 We need the formula for volume of a sphere based on the radius of the sphere:, where: radius of the sphere. In this problem, 6, so we get: 6 216 ft 3 Answer C 28. You want to design a cylindrical container for oatmeal that has a volume of 77. You also want the height of the container to be 2 times the radius. To the nearest tenth, what should the radius of the container be? A. 2.3 B. 2.9 C. 3.0 D. 3.1 We need the formula for volume of a cylinder based on the radius of a base and the height:, where: radius of a base, height of the cylinder. In this problem, 77, and 2, so we get: 77 2 2 77 2, and solving for gives: ~. inches Answer A 18 Page
29. Find the volume of the composite figure. Round your answer to the nearest tenth. A. 245 B. 441 C. 539 D. 735 The volume of the composite figure is the sum of the volumes of the square pyramid on top, and the rectangular prism on the bottom. For the square pyramid: 7 6 98 m 3 For the rectangular Prism: 779 441 m 3 Total volume is: 98 441 m 3 Answer C 30. What is the ratio of the volumes of the two cubes? The cubes have edges of lengths 3 inches and 12 inches. A. 1: 4 B. 1: 16 C. 1: 64 D. 1: 256 Some thoughts: If the ratio of side lengths is : in a pair of similar figures. Then: The measures of the angles in the two figures are the same. The ratio of any one dimensional measurements is :. The ratio of any two dimensional measurements (e.g., areas) is : The ratio of any three dimensional measurements (e.g., volumes) is : In this problem, the ratios of the lengths is 3: 12, which simplifies to 1: 4. Then, the ratio of the volumes is: 1 :4 : Answer C Note that the ratio of the surface areas is: 1 :4 1:16. Watch out for a question like this on the real final. 19 Page
31. Given mac = mbc and is a central angle, what is the value of and mbc? A. 18, mbc 136 B. 9.2, mbc 88 C. 18, mbc 88 D. 9.2, mbc 136 Let s add a couple of indicators (orange dashes) for measures that are equal. Now, let s work with the arc and angle shown. Recall that the measure of an inscribed angle is half the measure of the arc it subtends. This tells us that: 44 5 2 88 5 2 Solving for x, we get: (Note also that 52 88 ) To determine, note that: 360 (the whole circle) Since 88 and, we can convert the above equation to: 88 360 or 88 2 360 Solving for gives us. Answer A 32. What is the measure of angle x? A. 50 B. 35 C. 25 D. 5 An angle with a vertex outside the circle is half the difference of its subtended arcs. 60 10 Answer C The following pages contain information about circles, angles and their subtended arcs from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. 20 Page
Geometry Parts of Circles Center the middle of the circle. All points on the circle are the same distance from the center. Radius a line segment with one endpoint at the center and the other endpoint on the circle. The term radius is also used to refer to the distance from the center to the points on the circle. Diameter a line segment with endpoints on the circle that passes through the center. Arc a path along a circle. Minor Arc a path along the circle that is less than 180⁰. Major Arc a path along the circle that is greater than 180⁰. Semicircle a path along a circle that equals 180⁰. Sector a region inside a circle that is bounded by two radii and an arc. Secant Line a line that intersects the circle in exactly two points. Tangent Line a line that intersects the circle in exactly one point. Chord a line segment with endpoints on the circle that does not pass through the center. 21 Page
Geometry Angles and Circles Central Angle Inscribed Angle Vertex inside the circle Vertex outside the circle Tangent on one side Tangents on two sides 22 Page
33. What is the measure of RS? A. 54 B. 38 C. 32 D. 27 An angle with a vertex inside the circle is half the sum of its subtended arcs. 54 76 108 76 Answer C 34. What is the measure of the inscribed angle, if the ray is tangent to the circle? A. 140 B. 110 C. 70 D. 55 Arc measure is: This is a special case of the measure of an inscribed angle. Recall that the measure of an inscribed angle is half the measure of the arc it subtends. The arc we care about is the one for which the measure is not given in the problem. We must calculate it, as show above in magenta, to be. Then, 220 Answer B 23 Page
35. What is the length of the minor arc AB in the circle with a radius of 36? A. 9 B. 6 C. 3 36 D. 1.5 First, let s get the circumference of the whole circle: 236 72 Next, let s find out what part of the whole circle is represented by the arc. The arc is 30 out of a total of 360 in a complete circle. This is of the whole circle. Multiply these two values together to get the length of minor arc. 72 cm. Answer B 36. What is the area of a circular pool that has a circumference of 100? A. 10 C. 100 B. 50 D. 2500 First, find the radius of the pool from the circumference. 100 2 50 ft. Next, calculate the area of the circle from the formula: 50 ft 2 Answer D The following page contains information about arc length and sector area from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. 24 Page
Geometry Circle Lengths and Areas Circumference and Area is the circumference (i.e., the perimeter) of the circle. is the area of the circle. where: is the radius of the circle. Length of an Arc on a Circle A common problem in the geometry of circles is to measure the length of an arc on a circle. Definition: An arc is a segment along the circumference of a circle. where: AB is the measure (in degrees) of the arc. Note that this is also the measure of the central angle. is the circumference of the circle. Area of a Sector of a Circle Another common problem in the geometry of circles is to measure the area of a sector a circle. Definition: A sector is a region in a circle that is bounded by two radii and an arc of the circle. where: AB is the measure (in degrees) of the arc. Note that this is also the measure of the central angle. is the area of the circle. 25 Page
37. The diameter of a circular pizza pan is 18. Two-thirds of the pizza is eaten by your friends. What is the approximate area of the pizza pan that is covered by the remaining pizza? (Assume that the diameter of the pan and the diameter of the pizza are the same.) A. 170 C. 54 B. 85 D. 27 If your friends eat of a pizza, there is left for you. Note also that the radius of the pizza pan is The area of the whole pizza is: 9 81 in 2 9 inches. The portion of the pizza left for you, then, is: 81 27 ~ in2 Answer B 38. A sector of a circle has an area of 75 and an arc measure of 120. What is the radius of the circle? A. 10.6 C. 5 B. 21.2 D. 15 First, let s find out what part of the whole circle is represented by the arc. The arc is 120 out of a total of 360 in a complete circle. This is of the whole circle. So, the Area of the circle must be three times the size of the sector: 3 75 225. The radius of the circle can then be determined from the area formula: 225 225 cm Answer D 26 Page
39. Given the circle inscribed in the square with side length 12. What is the probability that the point lies inside the circle, if a point is chosen at random inside the square? A. C. B. 1 D. 4 4 The probability is the ratio of the shaded area to the total area. Note that the radius of the circle is half the length of the side of the square: 6 Answer D 40. Find the probability that a point chosen at random in the trapezoid shown lies in either of the shaded regions. Round your answer to the nearest hundredth. A. 0.61 B. 0.56 C. 0.39 D. 0.35 The probability is the ratio of the shaded area to the total area. I have added a couple of measurements to the above diagram to make the calculations easier. ~. Answer C The following page contains information regarding perimeters and areas of 2 dimensional shapes from the Geometry Handbook available at www.mathguy.us/bysubject/geometry.php. One interesting thing to keep in mind is that a square is also a kite. So if you are given the diagonal of a square and asked to find the area, use the formula 27 Page
Geometry Summary of Perimeter and Area Formulas 2D Shapes Shape Figure Perimeter Area Kite,, Trapezoid Parallelogram,,, b,b bases h height Rectangle, Rhombus, Square, Regular Polygon Circle 28 Page
41. A grab bag contains 7 football cards and 3 basketball cards. An experiment consists of taking one card out of the bag, then selecting another card. What is the probability of selecting a football card, replacing it, and then selecting a basketball card? A. 0.23 C. 0.21 B. 0.09 D. 0.49 Let s look at the probabilities for a draw. Notice that the probabilities don NOT change after the first draw because you replace the card chosen. Cards Probability Football Cards 7 7/ 10 0.7 Basketball Cards 3 3/10 0.3 Total Cards 10 Notice that the two draws are independent of each other. Two events are independent when neither one affects the other. When events are independent, we can multiply the probabilities of each event to get the overall probability.. 0. 7 0.3. Answer C 42. A bag contains hair ribbons for a spirit rally. The bag contains 5 black ribbons and 7 green ribbons. Lila selects a ribbon at random, then Jessica selects a ribbon at random from the remaining ribbons. Find the probability that both events and occur. A. :. :. 7 33 B. 35 144 C. 35 132 D. 5 22 Let s look at the probabilities for each draw. Notice that the probabilities change after the first draw because you do not replace the ribbon chosen. Lila Draw Jessica Draw Black Ribbons 5 4 Green Ribbons 7 7 Total Ribbons 12 11 Notice that the probabilities change between the two draws because they are not independent. We still multiply the two resulting probabilities. Answer C 29 Page
43. The table shows the distribution of male and female students and left- and right-handed students in the math club. Find the probability that a female student selected at random is left-handed. Which is the correct answer as a fraction in simplest form? Left-handed Right-handed Total Male 2 35 Female 6 36 42 A. 3 4 B. 1 7 C. 1 6 D. 6 79 We are given that the student is female, so we confine ourselves to that row. I added the total column so we can do our calculation: 6 42 Answer B 44. The table shows the distribution of the labor force in a city in the year 2000. Suppose that a worker is selected at random. Find the probability of randomly selecting a worker in the Industry field given that the worker is female. Which is the correct answer as a decimal rounded to the nearest thousandth? Agriculture Industry Services Total Male 3,132 25,056 50,112 Female 667 8,004 57,362 66,033 A. 0.141 B. 0.242 C. 0.121 D. 0.312 We are given that the worker is female, so we confine ourselves to that row. I added the total column so we can do our calculation: 8, 004 ~. 66,033 Answer C 30 Page
45. Events and are independent. Find the missing probability. (call it ) 0.3 0.06 A. 0.7 C. 0.2 B. 0.24 D. 0.3 If two events are independent, then. For this problem, 0.06 Solving for, we get... Answer 0.3 C 46. If 0.43 and 0. 89, find. A. 0.51 C. 0.11 B. 0.48 D. 0.38 The key formula to use here is: 0. 89 0.43 0.890.43 ~. Answer D 47. The sections on a spinner are numbered from 1 through 8. If the probability of landing on a given section is the same for all the sections, what is the probability of spinning a number less than 4 or greater than 7 in a single spin? A. B. The 1 2 1 8 successes are values 4or 7. So, 1,2,3,8. There are 4 possible successes out of a set of 8 total possibilities when spinning. The resulting probability of success is: C. D. 3 8 5 8 Answer A 31 Page
48. Given, a student constructed point as shown. Next the student will draw a circle with center and radius. Which statement is true and why? A. Circle will be inscribed in because point is the intersection of two angle bisectors of. B. Circle will be circumscribed about because point is the intersection of two angle bisectors of. C. Circle will be circumscribed about because point is the intersection of two perpendicular bisectors of sides of. D. Circle will be inscribed in because point is the intersection of two perpendicular bisectors of sides of. The center of the circle (O) in this problem is the intersection of angle bisectors of the triangle. This point of intersection is called the incenter of the triangle. The incenter is also the center of the circle inscribed in the triangle. The key words in the paragraph above are angle bisectors and inscribed. These are the words in Answer A. The following page contains information regarding other centers of triangles with which the student should be familiar. The page is from the Geometry Handbook available at: www.mathguy.us/bysubject/geometry.php. 32 Page
Geometry Centers of Triangles The following are all points which can be considered the center of a triangle. Centroid (Medians) The centroid is the intersection of the three medians of a triangle. A median is a line segment drawn from a vertex to the midpoint of the line opposite the vertex. The centroid is located 2/3 of the way from a vertex to the opposite side. That is, the distance from a vertex to the centroid is double the length from the centroid to the midpoint of the opposite line. The medians of a triangle create 6 inner triangles of equal area. Orthocenter (Altitudes) The orthocenter is the intersection of the three altitudes of a triangle. An altitude is a line segment drawn from a vertex to a point on the opposite side (extended, if necessary) that is perpendicular to that side. In an acute triangle, the orthocenter is inside the triangle. In a right triangle, the orthocenter is the right angle vertex. In an obtuse triangle, the orthocenter is outside the triangle. Circumcenter (Perpendicular Bisectors) The circumcenter is the intersection of the perpendicular bisectors of the three sides of the triangle. A perpendicular bisector is a line which both bisects the side and is perpendicular to the side. The circumcenter is also the center of the circle circumscribed about the triangle. In an acute triangle, the circumcenter is inside the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse. In an obtuse triangle, the circumcenter is outside the triangle. Euler Line: Interestingly, the centroid, orthocenter and circumcenter of a triangle are collinear (i.e., lie on the same line, which is called the Euler Line). Incenter (Angle Bisectors) The incenter is the intersection of the angle bisectors of the three angles of the triangle. An angle bisector cuts an angle into two congruent angles, each of which is half the measure of the original angle. The incenter is also the center of the circle inscribed in the triangle. 33 Page
49. Find the angle measures of. A. 34 54 146 126 B. 71 54 109 126 C. 35 134 145 54 D. 71 50 109 130 Opposite angles of a quadrilateral inscribed in a circle add to 180. So, we have: 2 3 3 7 180 5 10 180. Solving for, we get: 34. Then, substitute 34 into the measures of each angle to get our solution: 2 3 234 3 71 3 7 334 7 109 These two angles add to 180. 4 10 434 10 126 Finally, 180 180 126 54 Answer B 34 Page
50. Which circle is inscribed in the triangle? A. Circle B. Circle C. Circle D. All of the above Inscribed means inside the triangle. So that would be Circle. Answer A Geometry Semester 2 Instructional Material Answers 1. B 11. C 21. A 31. A 41. C 2. D 12. A 22. C 32. C 42. C 3. A 13. D 23. D 33. C 43. B 4. A 14. C 24. C 34. B 44. C 5. D 15. B 25. C 35. B 45. C 6. D 16. C 26. B 36. D 46. D 7. D 17. B 27. C 37. B 47. A 8. B 18. A 28. A 38. D 48. A 9. C 19. A 29. C 39. D 49. B 10. B 20. B 30. C 40. C 50. A 35 Page