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Ohio's State Tests Geometry

Congruence, similarity, circles, coordinates, area, and volume.

These questions are original. This is not an official Ohio's State Tests exam, and FreeExam is not affiliated with the Ohio Department of Education and Workforce.

  1. 1.A walking path at Caesar Creek meets a shelter path at a right angle. The legs of the right triangle measure 9 meters and 12 meters. How long is the hypotenuse?

  2. 2.On a map of the Ohio and Erie Canal towpath, a lock gate is plotted at (1, -4). The gate is translated 6 units right and 2 units up. What are the image coordinates?

  3. 3.A chart of Kelleys Island reflects a buoy across the x-axis. The buoy starts at (-3, 5). What are the image coordinates?

  4. 4.A kiosk corner on the Cleveland lakefront is drawn at (4, 1). The sketch rotates 90 degrees counterclockwise about the origin. Where does the corner go?

  5. 5.A garden corner in German Village is plotted at (-2, 6). The drawing is rotated 180 degrees about the origin. What are the image coordinates?

  6. 6.A trail sign in Cuyahoga Valley National Park is graphed at (8, -3) and then reflected across the y-axis. Which point is the image?

  7. 7.A mural grid in the Short North is dilated by a factor of 3 from the origin. A corner at (2, -1) moves to which point?

  8. 8.A footbridge over the Scioto River in Columbus runs from a point at (2, 4) to a point at (8, 10). What is the midpoint of that segment?

  9. 9.On a coordinate map of downtown Dayton, a fountain is at (0, 0) and a bike rack is at (3, 4). What is the distance between them?

  10. 10.A ramp beside the Maumee River in Toledo climbs from (1, 2) to (5, 10). What is the slope of the ramp?

  11. 11.A lane on a trail in Akron has slope 3/4. A second lane is parallel to the first. Which slope can the second lane have?

  12. 12.The edge of a plaza in Cincinnati has slope 2. A curb meets that edge at a right angle. What is the slope of the curb?

  13. 13.A circular fountain along the Cincinnati riverfront is centered at (3, -2) and has radius 5. Which equation matches the fountain?

  14. 14.A circle on a Sandusky harbor chart has center (1, 2) and radius 5. Which point lies on the circle?

  15. 15.A round stage on the Ohio Statehouse grounds has radius 4 feet. Using 3.14 for π, what is the area of the stage?

  16. 16.A circular flower bed in Wooster has radius 4 meters. Using 3.14 for π, what is the circumference?

  17. 17.A painted slice on a mural in Youngstown is a 90 degree arc of a circle with radius 4 inches. Using 3.14 for π, how long is the arc?

  18. 18.A stained-glass piece in Marietta is a 90 degree sector of a circle with radius 4 centimeters. Using 3.14 for π, what is the sector area?

  19. 19.In a circle drawn for a Chillicothe display, a central angle measures 80 degrees. An inscribed angle intercepts the same arc. What is the inscribed angle?

  20. 20.On a Put-in-Bay chart, PR is a diameter of a circle and Q is another point on the circle. What is the measure of angle PQR?

  21. 21.At a model of the Marblehead lighthouse, a tangent touches a circle and a radius meets the tangent at that same point. What angle do the radius and the tangent form?

  22. 22.From a point outside a circular pond in Granville, two tangent segments touch the pond. One tangent is 14 feet long. How long is the other tangent?

  23. 23.Two chords cross inside a circular window in Lancaster, Ohio. One chord is split into 3 centimeters and 8 centimeters. The other is split into 4 centimeters and an unknown length. What is the unknown length?

  24. 24.Two adjacent angles form a straight line along a curb in Springfield, Ohio. One angle measures 65 degrees. What is the measure of the other angle?

  25. 25.A carpenter in Troy, Ohio, cuts two boards so that the angles are complementary. One angle is 28 degrees. What is the other angle?

  26. 26.Two guy wires cross above a pole in Findlay and form vertical angles. One of the vertical angles measures 37 degrees. What is the opposite angle?

  27. 27.Two parallel rails in a Mansfield railyard are cut by a transversal. One corresponding angle measures 118 degrees. What is the matching corresponding angle?

  28. 28.Parallel streets in Shaker Heights are crossed by a transversal. An angle and its alternate interior angle are marked, and the given angle is 54 degrees. What is the alternate interior angle?

  29. 29.Parallel paths in Hueston Woods are cut by a transversal. One same-side interior angle is 70 degrees. What is the other same-side interior angle?

  30. 30.A triangular garden in Yellow Springs has two angles of 48 degrees and 67 degrees. What is the third angle?

  31. 31.At a trail junction in Mohican State Park, a triangle has remote interior angles of 35 degrees and 58 degrees. What is the exterior angle at the third vertex?

  32. 32.An isosceles triangle banner in Athens, Ohio, has a vertex angle of 44 degrees. The two base angles are congruent. What is each base angle?

  33. 33.A triangular medallion for a Circleville parade is equilateral. What is the measure of each interior angle?

  34. 34.A student in Delaware, Ohio, has sticks of lengths 5 inches, 7 inches, and 10 inches. Which statement is true?

  35. 35.Rods of lengths 2 meters, 3 meters, and 6 meters are laid beside the Great Miami River in Hamilton, Ohio. Which statement is true?

  36. 36.Which set of lengths can be the sides of a triangle on a school lot in Lima, Ohio?

  37. 37.A right triangle sail on a boat at Port Clinton has legs of 5 feet and 12 feet. How long is the hypotenuse?

  38. 38.A lot beside the Olentangy River in Worthington has side lengths 8 yards, 15 yards, and 17 yards. Which statement is true?

  39. 39.A garden triangle in Bexley has sides of 6 feet, 7 feet, and 10 feet. Which description fits?

  40. 40.An isosceles right triangle on a quilt in Holmes County has legs of 5 inches each. What is the hypotenuse?

  41. 41.A 30-60-90 roof triangle in Oberlin has a hypotenuse of 12 feet. What is the shorter leg, the side opposite the 30 degree angle?

  42. 42.The shorter leg of a 30-60-90 support in a covered bridge in Ashtabula County is 7 meters. How long is the longer leg?

  43. 43.A right triangle at a lookout in Hocking Hills has an acute angle with opposite side 8 meters and hypotenuse 17 meters. What is the sine of that angle?

  44. 44.For an acute angle of a right triangle beside Brandywine Falls, the adjacent side is 9 centimeters and the hypotenuse is 15 centimeters. What is the cosine of the angle?

  45. 45.A ladder leans against a vertical wall in Newark, Ohio. It reaches 8 feet up the wall, and its foot is 6 feet from the wall. What is the tangent of the angle between the ladder and the ground?

  46. 46.From a bluff at Clifton Gorge, a viewer looks up at a hawk. The angle of elevation is 30 degrees, and the line of sight is 40 meters. How far is the hawk above the viewer's eye level?

  47. 47.A cable runs straight from the top of a vertical pole to a ground anchor at John Bryan State Park. The cable is 18 feet long and makes a 60 degree angle with the ground. How far is the anchor from the base of the pole?

  48. 48.A ramp in Dayton rises at 45 degrees from horizontal ground. The horizontal run is 13 feet. What is the vertical rise?

  49. 49.Two triangles on a poster in Kent each have angles of 40 degrees and 60 degrees. Which conclusion must be true?

  50. 50.Similar triangular pennants in Bowling Green have corresponding sides of 4 inches and 10 inches. What is the scale factor from the smaller pennant to the larger one?

  51. 51.A square patio in Oxford, Ohio, has each side doubled to make a larger square patio. By what factor does the area change?

  52. 52.Two similar maps of Lake Erie use a scale factor of 3 from the smaller map to the larger map. A bay has area 5 square centimeters on the smaller map. What is the matching area on the larger map?

  53. 53.Two similar models of the Zanesville Y-bridge have a length scale factor of 2. The smaller model has volume 9 cubic inches. What is the volume of the larger model?

  54. 54.A meter stick and a tree in Logan cast shadows at the same time of day. The stick is 2 meters tall and casts a 3 meter shadow. The tree's shadow is 12 meters. How tall is the tree?

  55. 55.In a triangle drawn at Fort Ancient, a midsegment joins the midpoints of two sides and is parallel to the third side. The third side is 18 centimeters. How long is the midsegment?

  56. 56.Two triangles in a Steubenville bridge design have three pairs of corresponding sides congruent. Which congruence theorem applies?

  57. 57.A proof about two roof trusses in Massillon shows two sides and the included angle of one triangle congruent to the matching parts of the other triangle. Which theorem finishes the proof?

  58. 58.Two triangles on a diagram of the Newark Earthworks have two angles and the included side congruent to the corresponding parts of the other triangle. Which congruence statement is justified?

  59. 59.Two right triangles in a Perrysburg roof plan have congruent hypotenuses and one pair of congruent legs. Which theorem shows the triangles are congruent?

  60. 60.After two triangles in a Bellefontaine proof are shown congruent, a student claims the matching angles are congruent. Which reason supports that claim?

  61. 61.A rigid motion maps a trail figure in Wayne National Forest onto another figure. Which quantity must stay the same?

  62. 62.Which transformation of a Serpent Mound site plan is a rigid motion?

  63. 63.A drawing of Old Man's Cave is dilated by a factor of 2 about a point. Which statement is true?

  64. 64.A point on a Gallipolis river map starts at (2, -1). It is translated 3 units left and then 4 units up. What are the final coordinates?

  65. 65.A coordinate reflection across the line y = x sends a dock at (7, -2) on a Sandusky Bay chart to which point?

  66. 66.A corner of a parcel in Upper Arlington is at (-3, 4). It is rotated 90 degrees clockwise about the origin. Where does it land?

  67. 67.A segment on a bike map in Dublin, Ohio, runs from (0, 0) to (9, 6). A rest stop divides the segment in the ratio 1:2, with the shorter part starting at (0, 0). What are the rest stop's coordinates?

  68. 68.The endpoints of a footbridge in Lakewood are (-4, 6) and (2, -2). What is the midpoint?

  69. 69.How far apart are the points (1, -2) and (8, -2) on a horizontal street grid in Parma?

  70. 70.A surveyor in Medina, Ohio, plots a hydrant at (2, 1) and a valve at (5, 5). What is the distance between them?

  71. 71.A hillside trail near Ash Cave drops from (0, 8) to (4, 0). What is the slope?

  72. 72.A line on a coordinate grid in Canton, Ohio, passes through (5, 1) and (5, 9). What is the slope of the line?

  73. 73.A canal level in Roscoe Village is graphed through (-3, 4) and (6, 4). What is the slope of that line?

  74. 74.On a street grid in Columbus, which equation is parallel to y = 3x - 1 and passes through (0, 4)?

  75. 75.On a planning map in Westerville, which equation has a graph perpendicular to the line y = (1/2)x + 3?

  76. 76.A circular garden in Gahanna has equation (x + 1)^2 + (y - 4)^2 = 36. What are the center and the radius?

  77. 77.A circle centered at the origin has radius 10. How is the point (6, 8) related to the circle?

  78. 78.A round skating rink planned for Stow has radius 10 meters. Using 3.14 for π, what is the area of the rink?

  79. 79.A circular track around a field in Reynoldsburg has diameter 20 meters. Using 3.14 for π, what is the circumference?

  80. 80.A 60 degree sector of a circle with radius 6 feet is painted on a gym floor in Gahanna. Using 3.14 for π, what is the area of the sector?

  81. 81.A 120 degree arc of radius 3 meters appears on a sculpture in the Short North. Using 3.14 for π, what is the arc length?

  82. 82.A quadrilateral is inscribed in a circle on a diagram in Tiffin. One interior angle measures 74 degrees. What is the opposite interior angle?

  83. 83.A central angle in a courtyard diagram in University Circle measures 110 degrees. What is the measure of its intercepted arc?

  84. 84.In a circular window in Hudson, Ohio, a radius of 10 inches is perpendicular to a chord and meets the chord at its midpoint. Half of the chord is 6 inches. How far is the chord from the center?

  85. 85.From a point outside a circle in Chagrin Falls, a tangent of length 6 centimeters and a secant are drawn. The external part of the secant is 4 centimeters. What is the length of the whole secant?

  86. 86.A pentagonal gazebo in Mentor has five sides. What is the sum of its interior angles?

  87. 87.A regular hexagon tile is set into a floor in Lancaster, Ohio. What is the measure of each interior angle?

  88. 88.A regular octagon is printed on a poster in Fremont, Ohio. What is the measure of each exterior angle?

  89. 89.What is the sum of the interior angles of a hexagon drawn for a honeycomb mural in Medina, Ohio?

  90. 90.A regular polygon on a banner in Alliance, Ohio, has an exterior angle of 40 degrees. How many sides does the polygon have?

  91. 91.A parallelogram lot in Boardman, Ohio, has one side of 14 meters and an adjacent side of 9 meters. What is the perimeter?

  92. 92.One angle of a parallelogram window in Warren, Ohio, measures 68 degrees. What is the opposite angle?

  93. 93.One angle of a parallelogram frame in Niles, Ohio, is 125 degrees. What is a consecutive angle?

  94. 94.A rectangular stage in Youngstown is 8 meters by 15 meters. How long is a diagonal?

  95. 95.A rhombus kite flown over Buckeye Lake has diagonals of 10 feet and 16 feet. What is the area of the kite?

  96. 96.A trapezoidal garden in Pickerington has bases of 8 meters and 14 meters and height 5 meters. What is its area?

  97. 97.A triangular sail on Lake Erie near Vermilion has base 12 feet and height 7 feet. What is the area of the sail?

  98. 98.A parallelogram banner in Salem, Ohio, has base 11 inches and height 6 inches. What is its area?

  99. 99.A cylindrical cistern in Millersburg has radius 3 meters and height 10 meters. Using 3.14 for π, what is the volume?

  100. 100.A cylinder and a cone at a pottery studio in Zanesville have the same base and the same height. The cylinder holds 21 cubic units. How much does the cone hold?

  101. 101.A square pyramid model of a mound in Chillicothe has a base area of 36 square feet and a height of 4 feet. What is the volume?

  102. 102.A spherical tank ornament in Grove City has radius 3 feet. Using 3.14 for π, what is the volume of the sphere?

  103. 103.A rectangular storage box in Hilliard measures 4 feet by 5 feet by 6 feet. What is its volume?

  104. 104.A triangular prism tent frame in Mohican State Park has a triangular base of area 12 square feet and a length of 9 feet. What is the volume?

  105. 105.A cone-shaped pile of road salt in Marion, Ohio, has radius 3 meters and height 4 meters. Using 3.14 for π, what is the volume?

  106. 106.A closed cylindrical can in a Findlay factory has radius 2 centimeters and height 5 centimeters. Using 3.14 for π, what is the total surface area?

  107. 107.A stone block in a Flint Ridge display has volume 20 cubic centimeters and density 3 grams per cubic centimeter. What is its mass?

  108. 108.A plane cuts a right circular cone parallel to its base and does not pass through the apex. The model is displayed at Salt Fork State Park. What is the shape of the cross section?