[
  {
    "Question": "What does the Pythagorean theorem state?",
    "Answer": "A",
    "Explanation": "The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This can be expressed as a² + b² = c².",
    "PictureURL": "",
    "OptionA": "a² + b² = c²",
    "OptionB": "a² - b² = c²",
    "OptionC": "a + b = c",
    "OptionD": "a² + b² = 2c",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Pythagorean Theorem Basics",
    "Item": 1,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/basics"
  },
  {
    "Question": "In a right triangle, if one leg measures 3 units and the other leg measures 4 units, what is the length of the hypotenuse?",
    "Answer": "B",
    "Explanation": "Using the Pythagorean theorem: c² = 3² + 4² = 9 + 16 = 25. Therefore, c = √25 = 5 units.",
    "PictureURL": "",
    "OptionA": "3 units",
    "OptionB": "5 units",
    "OptionC": "7 units",
    "OptionD": "6 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Finding the Hypotenuse",
    "Item": 2,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/hypotenuse"
  },
  {
    "Question": "If the hypotenuse of a right triangle is 10 units and one leg is 6 units, what is the length of the other leg?",
    "Answer": "C",
    "Explanation": "Using the Pythagorean theorem: c² = a² + b². Here, 10² = 6² + b², which simplifies to 100 = 36 + b². Thus, b² = 64, and b = √64 = 8 units.",
    "PictureURL": "",
    "OptionA": "4 units",
    "OptionB": "5 units",
    "OptionC": "8 units",
    "OptionD": "12 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Finding a Leg",
    "Item": 3,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/leg"
  },
  {
    "Question": "Which of the following triangles can be classified as a right triangle?",
    "Answer": "D",
    "Explanation": "A triangle is classified as a right triangle if it satisfies the Pythagorean theorem. In this case, the triangle with sides 5, 12, and 13 satisfies 5² + 12² = 13².",
    "PictureURL": "",
    "OptionA": "3, 4, 5",
    "OptionB": "6, 8, 10",
    "OptionC": "7, 24, 25",
    "OptionD": "5, 12, 13",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Identifying Right Triangles",
    "Item": 4,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/right_triangle"
  },
  {
    "Question": "What is the distance between the points (3, 4) and (0, 0) in a Cartesian plane?",
    "Answer": "A",
    "Explanation": "The distance formula is derived from the Pythagorean theorem: d = √[(x2 - x1)² + (y2 - y1)²]. Here, d = √[(3 - 0)² + (4 - 0)²] = √(9 + 16) = √25 = 5.",
    "PictureURL": "",
    "OptionA": "5",
    "OptionB": "7",
    "OptionC": "4",
    "OptionD": "3",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Distance Calculation",
    "Item": 5,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/distance"
  },
  {
    "Question": "If a right triangle has legs of lengths 8 and 15, what is the area of the triangle?",
    "Answer": "B",
    "Explanation": "The area of a triangle is given by the formula: Area = 1/2 * base * height. Here, Area = 1/2 * 8 * 15 = 60 square units.",
    "PictureURL": "",
    "OptionA": "30 square units",
    "OptionB": "60 square units",
    "OptionC": "120 square units",
    "OptionD": "45 square units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Area of Right Triangle",
    "Item": 6,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/area"
  },
  {
    "Question": "In a right triangle, if the lengths of the legs are in the ratio 3:4, what is the ratio of the hypotenuse?",
    "Answer": "C",
    "Explanation": "If the legs are in the ratio 3:4, we can use the Pythagorean theorem. Let the legs be 3x and 4x. Then, hypotenuse = √[(3x)² + (4x)²] = √(9x² + 16x²) = √(25x²) = 5x. Thus, the ratio of the hypotenuse to the legs is 5:1.",
    "PictureURL": "",
    "OptionA": "3:4",
    "OptionB": "4:5",
    "OptionC": "5:1",
    "OptionD": "1:5",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Ratios in Right Triangles",
    "Item": 7,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/ratios"
  },
  {
    "Question": "What is the length of the diagonal of a rectangle with width 6 units and height 8 units?",
    "Answer": "D",
    "Explanation": "The diagonal can be found using the Pythagorean theorem: d = √(width² + height²) = √(6² + 8²) = √(36 + 64) = √100 = 10 units.",
    "PictureURL": "",
    "OptionA": "8 units",
    "OptionB": "6 units",
    "OptionC": "12 units",
    "OptionD": "10 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Diagonal of a Rectangle",
    "Item": 8,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/diagonal_rectangle"
  },
  {
    "Question": "Which of the following sets of lengths can form a right triangle?",
    "Answer": "A",
    "Explanation": "The lengths 7, 24, and 25 satisfy the Pythagorean theorem: 7² + 24² = 49 + 576 = 625 = 25².",
    "PictureURL": "",
    "OptionA": "7, 24, 25",
    "OptionB": "5, 5, 10",
    "OptionC": "8, 15, 20",
    "OptionD": "9, 12, 15",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Forming Right Triangles",
    "Item": 9,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/forming_right_triangles"
  },
  {
    "Question": "If a right triangle has a hypotenuse of 13 units and one leg of 5 units, what is the length of the other leg?",
    "Answer": "B",
    "Explanation": "Using the Pythagorean theorem: c² = a² + b². Here, 13² = 5² + b², which simplifies to 169 = 25 + b². Thus, b² = 144, and b = √144 = 12 units.",
    "PictureURL": "",
    "OptionA": "10 units",
    "OptionB": "12 units",
    "OptionC": "14 units",
    "OptionD": "15 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Finding the Other Leg",
    "Item": 10,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/other_leg"
  },
  {
    "Question": "What is the relationship between the sides of a 30-60-90 triangle?",
    "Answer": "C",
    "Explanation": "In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. The shortest side is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is the longest side.",
    "PictureURL": "",
    "OptionA": "1:1:1",
    "OptionB": "1:2:3",
    "OptionC": "1:√3:2",
    "OptionD": "1:1:√2",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "30-60-90 Triangle",
    "Item": 11,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/30-60-90"
  },
  {
    "Question": "What is the length of the hypotenuse of a right triangle with legs measuring 9 units and 12 units?",
    "Answer": "A",
    "Explanation": "Using the Pythagorean theorem: c² = 9² + 12² = 81 + 144 = 225. Therefore, c = √225 = 15 units.",
    "PictureURL": "",
    "OptionA": "15 units",
    "OptionB": "14 units",
    "OptionC": "13 units",
    "OptionD": "16 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Hypotenuse Length",
    "Item": 12,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/hypotenuse_length"
  },
  {
    "Question": "In a right triangle, if the lengths of the legs are 5 units and 12 units, what is the perimeter of the triangle?",
    "Answer": "B",
    "Explanation": "First, find the hypotenuse using the Pythagorean theorem: c² = 5² + 12² = 25 + 144 = 169, so c = 13 units. The perimeter is the sum of all sides: 5 + 12 + 13 = 30 units.",
    "PictureURL": "",
    "OptionA": "25 units",
    "OptionB": "30 units",
    "OptionC": "35 units",
    "OptionD": "40 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Perimeter of Right Triangle",
    "Item": 13,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/perimeter"
  },
  {
    "Question": "What is the length of the altitude to the hypotenuse of a right triangle with legs of lengths 6 and 8?",
    "Answer": "C",
    "Explanation": "The area of the triangle can be calculated as (1/2) * base * height = (1/2) * 6 * 8 = 24. The hypotenuse is 10 units. The altitude can be found using the area formula: Area = (1/2) * hypotenuse * altitude. Thus, 24 = (1/2) * 10 * altitude, leading to altitude = 4.8 units.",
    "PictureURL": "",
    "OptionA": "3 units",
    "OptionB": "5 units",
    "OptionC": "4.8 units",
    "OptionD": "6 units",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Altitude to Hypotenuse",
    "Item": 14,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/altitude"
  },
  {
    "Question": "Which of the following is NOT a property of right triangles?",
    "Answer": "D",
    "Explanation": "Right triangles have one angle that is exactly 90 degrees, and the sum of the angles in any triangle is 180 degrees. However, a right triangle cannot have all angles equal, as that would make it an equilateral triangle.",
    "PictureURL": "",
    "OptionA": "One angle is 90 degrees",
    "OptionB": "The sum of angles is 180 degrees",
    "OptionC": "The sides follow the Pythagorean theorem",
    "OptionD": "All angles are equal",
    "OptionE": "",
    "OptionF": "",
    "OptionG": "",
    "TestName": "Geometry Practice Test",
    "Content Type": "Mathematics",
    "Title": "Properties of Right Triangles",
    "Item": 15,
    "Type": "multiple choice",
    "Path": "geometry/pythagorean_theorem/properties"
  }
]