data_source stringclasses 1
value | prompt listlengths 1 1 | images images listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nConsider the points $A$, $B$, and $C$. At which point, $D$, do the lines formed by $BA$ and $AC$ intersect?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-3, -1)",
"style": "rule"
} | {
"answer": "(-3, -1)",
"difficulty": 1,
"has_image": true,
"id": 244102,
"index": 0,
"question": "Consider the points $A(-3,-1)$, $B(-1,-2)$, and $C(-2,6)$. At which point, $D$, do the lines formed by $BA$ and $AC$ intersect?\n Solve this question step by step and put your final answer in \\boxed{}.",
"s... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the amplitude, period, and frequency of the simple harmonic motion.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "5, 1, 1",
"style": "rule"
} | {
"answer": "5, 1, 1",
"difficulty": 1,
"has_image": true,
"id": 244241,
"index": 1,
"question": "Find the amplitude, period, and frequency of the simple harmonic motion described by the equation \\( y = 5 \\cos 2\\pi t \\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"spl... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the geometrical locus represented by the equation \\( x^2 + y^2 - 2x - 3 = 0 \\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(1, 0), 2",
"style": "rule"
} | {
"answer": "(1, 0), 2",
"difficulty": 1,
"has_image": true,
"id": 244256,
"index": 2,
"question": "Find the geometrical locus represented by the equation \\( x^2 + y^2 - 2x - 3 = 0 \\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the measures of \\( x \\) and \\( y \\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "104°, 38°",
"style": "rule"
} | {
"answer": "104°, 38°",
"difficulty": 1,
"has_image": true,
"id": 244336,
"index": 3,
"question": "In an isosceles triangle, one of the base angles is 38°. Find the measures of the vertex angle \\( x \\) and the other base angle \\( y \\).\n Solve this question step by step and put your final answer in \\b... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the distance?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{1095}",
"style": "rule"
} | {
"answer": "\\sqrt{1095}",
"difficulty": 1,
"has_image": true,
"id": 244357,
"index": 4,
"question": "What is the distance from the origin $(0,0)$ to $(32,\\sqrt{71})$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the distance, in units, between point $A$ and point $B$? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{282}",
"style": "rule"
} | {
"answer": "\\sqrt{282}",
"difficulty": 1,
"has_image": true,
"id": 244427,
"index": 5,
"question": "What is the distance, in units, between point $A = ( -7, -6, -5)$ and point $B = ( 9, -5, 0)$? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nJohn draws 20 pictures. How many square feet total does John draw?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "55.325",
"style": "rule"
} | {
"answer": "55.325",
"difficulty": 1,
"has_image": true,
"id": 244691,
"index": 6,
"question": "John draws 20 pictures. The first 5 are rectangles, each 1 foot long and 1 foot wide. The next 10 are triangles, each 1 foot long on the base and 3 feet tall. The remaining 5 are circles with diameter 3 feet. ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe circumference of a circle is equal to some number times pi. What is the circumference of the circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "72\\pi",
"style": "rule"
} | {
"answer": "72\\pi",
"difficulty": 1,
"has_image": true,
"id": 244981,
"index": 7,
"question": "The circumference of a circle is equal to some number times pi. The radius of this circle is 36. What is the circumference of the circle?\n Solve this question step by step and put your final answer in \\boxed{}... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the center and the radius of the circle given by $(x+2)^2 + (y+3)^2 = 2$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "C : (-2, -3), \\sqrt{2}",
"style": "rule"
} | {
"answer": "C : (-2, -3), \\sqrt{2}",
"difficulty": 1,
"has_image": true,
"id": 245000,
"index": 8,
"question": "Find the center and the radius of the circle given by $(x+2)^2 + (y+3)^2 = 2$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe equation $x^2 + y^2 - 6x = 0$ represents a circle. The coordinates of the center of the circle are ______; the radius is ______.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3",
"style": "rule"
} | {
"answer": "3",
"difficulty": 1,
"has_image": true,
"id": 245277,
"index": 9,
"question": "The equation $x^2 + y^2 - 6x = 0$ represents a circle. The coordinates of the center of the circle are \\_\\_\\_\\_\\_\\_; the radius is \\_\\_\\_\\_\\_\\_.\n Solve this question step by step and put your final answe... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nConvert the polar coordinate equation $\\rho=2$ into rectangular coordinate equation.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "x^2+y^2=4",
"style": "rule"
} | {
"answer": "x^2+y^2=4",
"difficulty": 1,
"has_image": true,
"id": 245293,
"index": 10,
"question": "Convert the polar coordinate equation $\\rho=2$ into rectangular coordinate equation.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind an expression for the area of the triangle in terms of a, b, and c, using Heron's formula.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{\\left( \\frac{a + b + c}{2} \\right)\\left( \\frac{a + b + c}{2} - a \\right)\\left( \\frac{a + b + c}{2} - b \\right)\\left( \\frac{a + b + c}{2} - c \\right)}",
"style": "rule"
} | {
"answer": "\\sqrt{\\left( \\frac{a + b + c}{2} \\right)\\left( \\frac{a + b + c}{2} - a \\right)\\left( \\frac{a + b + c}{2} - b \\right)\\left( \\frac{a + b + c}{2} - c \\right)}",
"difficulty": 1,
"has_image": true,
"id": 245364,
"index": 11,
"question": "Let the sides of a triangle be a, b, and c. Find... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIf the area of the first triangle is approximately $9 \\times 10^{-27}$ square units, what is the area, in square units, of the second triangle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "9r^2 \\times 10^{-27}",
"style": "rule"
} | {
"answer": "9r^2 \\times 10^{-27}",
"difficulty": 1,
"has_image": true,
"id": 245430,
"index": 12,
"question": "Mesh work is done on the vertices of a possible triangle to define a new set of vertices forming a triangle similar to the first one, such that each of the lengths of their sides are proportioned... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA trapezoid has parallel bases. What is the side length of a square?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{184}",
"style": "rule"
} | {
"answer": "\\sqrt{184}",
"difficulty": 1,
"has_image": true,
"id": 245476,
"index": 13,
"question": "A trapezoid has parallel bases of 12 and 14 units, forming a 45 degree angle at one base. The trapezoid has an area of 184 square units. What is the side length of a square whose area is the same as that o... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA hemisphere has a volume of $4000\\pi$ cubic units. What is the height of the hemisphere?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt[3]{6000}",
"style": "rule"
} | {
"answer": "\\sqrt[3]{6000}",
"difficulty": 1,
"has_image": true,
"id": 245608,
"index": 14,
"question": "A hemisphere has a volume of $4000\\pi$ cubic units. What is the height of the hemisphere?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the area of a circle inscribed in a square? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi s^2}{4}",
"style": "rule"
} | {
"answer": "\\frac{\\pi s^2}{4}",
"difficulty": 1,
"has_image": true,
"id": 245745,
"index": 15,
"question": "What is the area of a circle inscribed in a square? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subse... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\na. Find the volumes of the baseball and softball. \nb. Find the surface areas of the baseball and softball.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{243}{2\\pi^2}, \\frac{288}{\\pi^2}, \\frac{81}{\\pi}, \\frac{144}{\\pi}",
"style": "rule"
} | {
"answer": "\\frac{243}{2\\pi^2}, \\frac{288}{\\pi^2}, \\frac{81}{\\pi}, \\frac{144}{\\pi}",
"difficulty": 1,
"has_image": true,
"id": 245891,
"index": 16,
"question": "A baseball has a great circle circumference of 9 inches, and a softball has a great circle circumference of 12 inches. \na. Find the volu... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the coordinates of the point symmetric to the point (1, -2, 3) with respect to the coordinate plane yOz.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-1, -2, 3)",
"style": "rule"
} | {
"answer": "(-1, -2, 3)",
"difficulty": 1,
"has_image": true,
"id": 245951,
"index": 17,
"question": "In the space rectangular coordinate system O-xyz, find the coordinates of the point symmetric to the point (1, -2, 3) with respect to the coordinate plane yOz.\n Solve this question step by step and put yo... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the height of the cylinder, in units? Express the height in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{11.7667}{\\pi}",
"style": "rule"
} | {
"answer": "\\frac{11.7667}{\\pi}",
"difficulty": 1,
"has_image": true,
"id": 246011,
"index": 18,
"question": "The volume of a cylinder is 105.9 cubic units. If the radius of the cylinder is 3 units, what is the height of the cylinder, in units? Express the height in terms of $\\pi$.\n Solve this question... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nDetermine the coordinates of the center and the radius of the circle described by the equation $(x-2)^2+y^2=4$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "B",
"style": "rule"
} | {
"answer": "B",
"difficulty": 1,
"has_image": true,
"id": 246064,
"index": 19,
"question": "Determine the coordinates of the center and the radius of the circle described by the equation $(x-2)^2+y^2=4$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"sub... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the equation of the set of all points that have a distance of 5 units from the y-axis.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "x = 5 or x = -5",
"style": "rule"
} | {
"answer": "x = 5 or x = -5",
"difficulty": 1,
"has_image": true,
"id": 246550,
"index": 20,
"question": "Find the equation of the set of all points that have a distance of 5 units from the y-axis.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the coordinate of point B?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(0, 2) or (6, 2)",
"style": "rule"
} | {
"answer": "(0, 2) or (6, 2)",
"difficulty": 1,
"has_image": true,
"id": 246761,
"index": 21,
"question": "In the Cartesian coordinate system, line segment AB is parallel to the x-axis and has a length of 3. If point A is at (3, 2), what is the coordinate of point B?\n Solve this question step by step and ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nMalia has an ice cream cone. What is the volume, in cubic cm, of Malia's ice cream cone? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "168.75\\pi",
"style": "rule"
} | {
"answer": "168.75\\pi",
"difficulty": 1,
"has_image": true,
"id": 246816,
"index": 22,
"question": "Malia has an ice cream cone that is 15 cm wide at its top, 15 cm wide at the bottom, and has a height of 9 cm. What is the volume, in cubic cm, of Malia's ice cream cone? Express your answer in terms of $\\... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIn Calabasas, a young aspiring musician is eager to start learning to play a ukulele. The child's parents decide to build a custom music stand for the ukulele. What is the length of the hypotenuse where the ukulele will rest? Additionally, how much ribbon is needed?\n Solve this question ... | math | {
"ground_truth": "20, 48",
"style": "rule"
} | {
"answer": "20, 48",
"difficulty": 1,
"has_image": true,
"id": 246842,
"index": 23,
"question": "In Calabasas, a young aspiring musician is eager to start learning to play a ukulele. The child's parents decide to build a custom music stand for the ukulele. The music stand will be in the shape of a right tr... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the linear speed \\( v \\) in inches per second, the angular speed in revolutions per minute (RPM), and the angular speed in radians per second?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2\\pi}{3} \\text{ inches/sec}, 2.5 \\text{ RPM}, \\frac{\\pi}{12} \\text{ radians/sec}",
"style": "rule"
} | {
"answer": "\\frac{2\\pi}{3} \\text{ inches/sec}, 2.5 \\text{ RPM}, \\frac{\\pi}{12} \\text{ radians/sec}",
"difficulty": 1,
"has_image": true,
"id": 246917,
"index": 24,
"question": "A wheel of radius 8 inches is rotating at 15 degrees per second. What is the linear speed \\( v \\) in inches per second, t... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe measure of angle $\\alpha$ in radians can be expressed as \\_\\_\\_\\_\\_\\_.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{4} + k\\pi",
"style": "rule"
} | {
"answer": "\\frac{\\pi}{4} + k\\pi",
"difficulty": 1,
"has_image": true,
"id": 247074,
"index": 25,
"question": "If the terminal side of angle $\\alpha$ lies on the line $y=x$, then the measure of angle $\\alpha$ in radians can be expressed as \\_\\_\\_\\_\\_\\_.\n Solve this question step by step and put... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the coordinates of the focus $F$ of the parabola.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(2, 0)",
"style": "rule"
} | {
"answer": "(2, 0)",
"difficulty": 1,
"has_image": true,
"id": 247438,
"index": 26,
"question": "Find the coordinates of the focus $F$ of the parabola $y^2 = 8x$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA shape is folded along a straight line, and the parts on both sides of the line can coincide with each other. These two parts are necessarily (is/are not) congruent. This shape is called a __________ shape, and this line is called a __________. The points that coincide after folding are ... | math | {
"ground_truth": "is, symmetrical, axis of symmetry, symmetrical points",
"style": "rule"
} | {
"answer": "is, symmetrical, axis of symmetry, symmetrical points",
"difficulty": 1,
"has_image": true,
"id": 248533,
"index": 27,
"question": "A shape is folded along a straight line, and the parts on both sides of the line can coincide with each other. These two parts are necessarily (is/are not) congrue... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA wheel rolls without slipping on the ground. How many complete revolutions does it make when it travels the distance shown along the ground?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{\\pi}",
"style": "rule"
} | {
"answer": "\\frac{3}{\\pi}",
"difficulty": 1,
"has_image": true,
"id": 248649,
"index": 28,
"question": "A wheel of radius 1 meter rolls without slipping on the ground. How many complete revolutions does it make when it travels 6 meters along the ground?\n Solve this question step by step and put your fin... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nEvaluate: \n(a) \\(\\sin \\theta\\); \n(b) \\(\\cos 2\\theta\\); \n(c) \\(\\sin 2\\theta\\); \n(d) \\(\\cos \\frac{\\theta}{2}\\); \n(e) \\(\\sin \\frac{\\theta}{2}\\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule"
} | {
"answer": "\\dfrac{1}{2}",
"difficulty": 1,
"has_image": true,
"id": 248917,
"index": 29,
"question": "If \\(\\theta\\) is an acute angle and \\(\\cos \\theta = \\frac{1}{2}\\), evaluate: \n(a) \\(\\sin \\theta\\); \n(b) \\(\\cos 2\\theta\\); \n(c) \\(\\sin 2\\theta\\); \n(d) \\(\\cos \\frac{\\theta}{... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the general solution for θ in the equation $\\sin \\theta = \\frac{\\sqrt{3}}{2}$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\theta = \\frac{\\pi}{3} + 2\\pi k \\text{ or } \\theta = \\frac{2\\pi}{3} + 2\\pi k \\text{ for any integer } k",
"style": "rule"
} | {
"answer": "\\theta = \\frac{\\pi}{3} + 2\\pi k \\text{ or } \\theta = \\frac{2\\pi}{3} + 2\\pi k \\text{ for any integer } k",
"difficulty": 1,
"has_image": true,
"id": 250249,
"index": 30,
"question": "What is the general solution for θ in the equation $\\sin \\theta = \\frac{\\sqrt{3}}{2}$?\n Solve this... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the base.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "15",
"style": "rule"
} | {
"answer": "15",
"difficulty": 1,
"has_image": true,
"id": 250497,
"index": 31,
"question": "Find the base provided the distance $AB=21m$, $AC=14m$ and $BC=15m$\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nGiven the polar coordinates $\\left( 2 \\sqrt{3}, \\frac{2 \\pi}{3} \\right)$, convert the rectangular coordinates back to polar coordinates, but this time, the angle is given in degrees instead of radians.\n Solve this question step by step and put your final answer in \\boxed{}.",
"... | math | {
"ground_truth": "\\left( 2 \\sqrt{3}, 120^{\\circ} \\right)",
"style": "rule"
} | {
"answer": "\\left( 2 \\sqrt{3}, 120^{\\circ} \\right)",
"difficulty": 1,
"has_image": true,
"id": 250541,
"index": 32,
"question": "Given the polar coordinates $\\left( 2 \\sqrt{3}, \\frac{2 \\pi}{3} \\right)$, convert the rectangular coordinates back to polar coordinates, but this time, the angle is give... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the coordinates of the point symmetric to $P$ with respect to the $yOz$ plane.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-1, -2, 3)",
"style": "rule"
} | {
"answer": "(-1, -2, 3)",
"difficulty": 1,
"has_image": true,
"id": 250591,
"index": 33,
"question": "In the three-dimensional Cartesian coordinate system $O$-$xyz$, find the coordinates of the point symmetric to the point $(1, -2, 3)$ with respect to the $yOz$ plane.\n Solve this question step by step and... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the equation of the common chord between two circles?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "S_1 - S_2 = 0",
"style": "rule"
} | {
"answer": "S_1 - S_2 = 0",
"difficulty": 1,
"has_image": true,
"id": 250983,
"index": 34,
"question": "What is the equation of the common chord between two circles \\(S_1 = x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0\\) and \\(S_2 = x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0\\)?\n Solve this question step by step and pu... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the original distance? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{265}",
"style": "rule"
} | {
"answer": "\\sqrt{265}",
"difficulty": 1,
"has_image": true,
"id": 251416,
"index": 35,
"question": "What is the original distance between the points (0,3) and (16,6)? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "tra... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the diagonal of the rectangle in meters?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{17}",
"style": "rule"
} | {
"answer": "\\sqrt{17}",
"difficulty": 1,
"has_image": true,
"id": 251652,
"index": 36,
"question": "Side lengths of a rectangle are 3 m and sqrt(8) m. What is the diagonal in meters?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nCompute the value of $x$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"answer": "\\frac{\\pi}{4}",
"difficulty": 1,
"has_image": true,
"id": 251755,
"index": 37,
"question": "Compute the value of $x$ in the equation $\\sin x = \\cos x$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWrite the equation in standard form of the circle.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x + 4)^2 + (y + 3)^2 = 52",
"style": "rule"
} | {
"answer": "(x + 4)^2 + (y + 3)^2 = 52",
"difficulty": 1,
"has_image": true,
"id": 251787,
"index": 38,
"question": "Write the equation in standard form of a circle whose center is (-4, -3) and that passes through (0, 3).\n Solve this question step by step and put your final answer in \\boxed{}.",
"split... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the distance in radians? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3\\pi}{2}",
"style": "rule"
} | {
"answer": "\\frac{3\\pi}{2}",
"difficulty": 1,
"has_image": true,
"id": 251843,
"index": 39,
"question": "What is the distance in radians of a quarter circle of radius of 3? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "tra... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"answer": "\\sqrt{5}",
"difficulty": 1,
"has_image": true,
"id": 251855,
"index": 40,
"question": "Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates: \\( x^2 + y^2 = 5 \\).\n Solve this question step by step and put your final answer in \\boxed{... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the length of one side of the square?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "2.29 \\, \\text{m}",
"style": "rule"
} | {
"answer": "2.29 \\, \\text{m}",
"difficulty": 1,
"has_image": true,
"id": 252107,
"index": 41,
"question": "If the area of a square is given by the equation \\( A = 5.25 \\, \\text{m}^2 \\), what is the length of one side of the square?\n Solve this question step by step and put your final answer in \\box... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the standard form of the equation of the circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x + 7)^2 + (y - 4)^2 = 49",
"style": "rule"
} | {
"answer": "(x + 7)^2 + (y - 4)^2 = 49",
"difficulty": 1,
"has_image": true,
"id": 252541,
"index": 42,
"question": "What is the standard form of the equation of the circle with:\n\ncenter at $(-7,4)$ and tangent to the $y$-axis?\n Solve this question step by step and put your final answer in \\boxed{}.",
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIn terms of $\\pi$, what is the area of the circle defined by the equation $3x^2 + 3y^2 - 9x + 12y + 27 = 0$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{11}{4}\\pi",
"style": "rule"
} | {
"answer": "\\frac{11}{4}\\pi",
"difficulty": 1,
"has_image": true,
"id": 253798,
"index": 43,
"question": "In terms of $\\pi$, what is the area of the circle defined by the equation $3x^2 + 3y^2 - 9x + 12y + 27 = 0$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the center of the ellipse with equation $x^2+4y^2=4$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(0, 0)",
"style": "rule"
} | {
"answer": "(0, 0)",
"difficulty": 1,
"has_image": true,
"id": 253885,
"index": 44,
"question": "What is the center of the ellipse with equation $x^2+4y^2=4$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nEvaluate $\\sec(852^\\circ)$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\sec(48^\\circ)",
"style": "rule"
} | {
"answer": "-\\sec(48^\\circ)",
"difficulty": 1,
"has_image": true,
"id": 254306,
"index": 45,
"question": "Evaluate $\\sec(852^\\circ)$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the center and the radius of the circle with the equation $4x^2 - 8x + 4y^2 - 16y + 20 = 0.$\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(1, 2), 0",
"style": "rule"
} | {
"answer": "(1, 2), 0",
"difficulty": 1,
"has_image": true,
"id": 254390,
"index": 46,
"question": "Find the center and the radius of the circle with the equation $4x^2 - 8x + 4y^2 - 16y + 20 = 0.$\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nAs a music journalist, you are designing a unique album cover for an upcoming hip-hop artist. The cover is a rectangle. You decide to include a series of geometric shapes that represent the artist's journey, including a central circular logo and four smaller triangular sections, each with... | math | {
"ground_truth": "180 - 16\\pi, \\frac{2\\pi}{27}",
"style": "rule"
} | {
"answer": "180 - 16\\pi, \\frac{2\\pi}{27}",
"difficulty": 1,
"has_image": true,
"id": 254391,
"index": 47,
"question": "As a music journalist, you are designing a unique album cover for an upcoming hip-hop artist. The cover is a rectangle with dimensions 12 inches by 18 inches. You decide to include a se... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the area of triangle $ABC$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "4(\\sqrt{57} + \\sqrt{17})",
"style": "rule"
} | {
"answer": "4(\\sqrt{57} + \\sqrt{17})",
"difficulty": 1,
"has_image": true,
"id": 254821,
"index": 48,
"question": "In triangle $ABC$, $AB = 11$, $AC = 9$, and the length of altitude $AN$ is 8. Find the area of triangle $ABC$.\n Solve this question step by step and put your final answer in \\boxed{}.",
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nHow can you create the circumcentre of a triangle using paper folding? Include the steps to form the perpendicular bisectors and identify the circumcentre.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "The circumcentre is found by folding the perpendicular bisectors of the triangle's sides and identifying their point of intersection.",
"style": "rule"
} | {
"answer": "The circumcentre is found by folding the perpendicular bisectors of the triangle's sides and identifying their point of intersection.",
"difficulty": 1,
"has_image": true,
"id": 254825,
"index": 49,
"question": "How can you create the circumcentre of a triangle using paper folding? Include the ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIn polar coordinates, the point is the same as what other point in a counterclockwise direction?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(0, \\frac{29\\pi}{12})",
"style": "rule"
} | {
"answer": "(0, \\frac{29\\pi}{12})",
"difficulty": 1,
"has_image": true,
"id": 254981,
"index": 50,
"question": "In polar coordinates, the point $\\left( 0, \\frac{5 \\pi}{12} \\right)$ is the same as what other point in a counterclockwise direction?\n Solve this question step by step and put your final a... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nDetermine the area of the sector of a circle. Express your answer in square centimeters in terms of pi. (Use 3.14 for $\\pi$.)\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{53}{40} \\pi",
"style": "rule"
} | {
"answer": "\\frac{53}{40} \\pi",
"difficulty": 1,
"has_image": true,
"id": 255232,
"index": 52,
"question": "Determine the area of the sector of a circle with a radius of 3 cm and a central angle of 53 degrees. Express your answer in square centimeters in terms of pi. (Use 3.14 for $\\pi$.)\n Solve this q... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWrite the equation of the circle.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x-3)^{2}+(y-1)^{2}=10",
"style": "rule"
} | {
"answer": "(x-3)^{2}+(y-1)^{2}=10",
"difficulty": 1,
"has_image": true,
"id": 255274,
"index": 53,
"question": "A circle with center at the point \\( M(3, 1) \\) passes through the origin. Write the equation of the circle.\n Solve this question step by step and put your final answer in \\boxed{}.",
"spl... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the distance between S and U? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{274}",
"style": "rule"
} | {
"answer": "\\sqrt{274}",
"difficulty": 1,
"has_image": true,
"id": 255424,
"index": 54,
"question": "point S(0,0) and U(-7,15), what is the distance between S and U? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA sheets of graph paper is requ red. Label the points on an x-axis and y-axis, then put these labels on the graph paper. For each ordered pair $(x,y)$ shown in the table below, plot a dot at the corresponding point.\n Solve this question step by step and put your final answer in \\boxed{}... | math | {
"ground_truth": "(0, -1), (-1, -4), (1, 2), (-2, -2), (-3, -5)",
"style": "rule"
} | {
"answer": "(0, -1), (-1, -4), (1, 2), (-2, -2), (-3, -5)",
"difficulty": 1,
"has_image": true,
"id": 255470,
"index": 55,
"question": "A sheets of graph paper is requ red. Label the points on an x-axis and y-axis from $-5$ to $5,$ then put these labels on the graph paper. For each ordered pair $(x,y)$ sho... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nPlot the polar point \\((-2, \\frac{\\pi}{4})\\). Explain the steps involved in plotting this point.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "The point \\((-2, \\frac{\\pi}{4})\\) is plotted at \\((2, \\frac{5\\pi}{4})\\) in the third quadrant.",
"style": "rule"
} | {
"answer": "The point \\((-2, \\frac{\\pi}{4})\\) is plotted at \\((2, \\frac{5\\pi}{4})\\) in the third quadrant.",
"difficulty": 1,
"has_image": true,
"id": 255881,
"index": 56,
"question": "Plot the polar point \\((-2, \\frac{\\pi}{4})\\). Explain the steps involved in plotting this point.\n Solve this ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nCalculate the radius of the sphere.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "7.8",
"style": "rule"
} | {
"answer": "7.8",
"difficulty": 1,
"has_image": true,
"id": 256010,
"index": 57,
"question": "The surface area of a sphere is 760.0 square centimeters. Calculate the radius of the sphere.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-130... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the argument of the complex number \\(3 - j4\\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\tan^{-1}\\left(\\frac{-4}{3}\\right)",
"style": "rule"
} | {
"answer": "\\tan^{-1}\\left(\\frac{-4}{3}\\right)",
"difficulty": 1,
"has_image": true,
"id": 256148,
"index": 58,
"question": "Find the argument of the complex number \\(3 - j4\\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the radius of the semi-circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "21.98",
"style": "rule"
} | {
"answer": "21.98",
"difficulty": 1,
"has_image": true,
"id": 256192,
"index": 59,
"question": "The perimeter of a semi-circle is 113 cm. What is the radius of the semi-circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nLet \\( F \\) be the mapping defined by \\( F(t,u) = (\\cos t, \\sin t, u) \\). Describe geometrically the image of the \\((t,u)\\)-plane under \\( F \\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "A circular cylinder of radius 1 with the z-axis as its central axis",
"style": "rule"
} | {
"answer": "A circular cylinder of radius 1 with the z-axis as its central axis",
"difficulty": 1,
"has_image": true,
"id": 256390,
"index": 60,
"question": "Let \\( F \\) be the mapping defined by \\( F(t,u) = (\\cos t, \\sin t, u) \\). Describe geometrically the image of the \\((t,u)\\)-plane under \\( F... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nDerive the equation of the line, using the fact that the gradient of \\( P_1P \\) is equal to the gradient of \\( P_1P_2 \\). Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(y_2 - y_1)x - (x_2 - x_1)y + (x_2y_1 - x_1y_2) = 0",
"style": "rule"
} | {
"answer": "(y_2 - y_1)x - (x_2 - x_1)y + (x_2y_1 - x_1y_2) = 0",
"difficulty": 1,
"has_image": true,
"id": 257281,
"index": 61,
"question": "Derive the equation of the line passing through two points \\( P_1(x_1, y_1) \\) and \\( P_2(x_2, y_2) \\), using the fact that if a third point \\( P(x, y) \\) is c... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the equation of the circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x - 4)^2 + y^2 = 40",
"style": "rule"
} | {
"answer": "(x - 4)^2 + y^2 = 40",
"difficulty": 1,
"has_image": true,
"id": 257551,
"index": 62,
"question": "What is the equation of the circle with a diameter with endpoints $(-2,2)$ and $(10,-2)$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the equation of the circle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(x + 4)^2 + (y + 3)^2 = 180",
"style": "rule"
} | {
"answer": "(x + 4)^2 + (y + 3)^2 = 180",
"difficulty": 1,
"has_image": true,
"id": 257669,
"index": 63,
"question": "What is the equation of the circle with center $(-4,-3)$ and passes through the point $(8,-9)$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "trai... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe symmetric point of the point (2, 3, 4) with respect to the yoz plane is ______.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-2, 3, 4)",
"style": "rule"
} | {
"answer": "(-2, 3, 4)",
"difficulty": 1,
"has_image": true,
"id": 257897,
"index": 64,
"question": "The symmetric point of the point (2, 3, 4) with respect to the yoz plane is \\_\\_\\_\\_\\_\\_.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe symmetric point of point A with respect to the y-axis is.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-2, -3)",
"style": "rule"
} | {
"answer": "(-2, -3)",
"difficulty": 1,
"has_image": true,
"id": 257998,
"index": 65,
"question": "The symmetric point of point A(2, -3) with respect to the y-axis is.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind $\\theta$, in degrees, for $0^\\circ \\le \\theta \\le 180^\\circ,$ for which $\\csc \\theta = 2.$\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "30^\\circ \\text{ and } 150^\\circ",
"style": "rule"
} | {
"answer": "30^\\circ \\text{ and } 150^\\circ",
"difficulty": 1,
"has_image": true,
"id": 258062,
"index": 66,
"question": "Find $\\theta$, in degrees, for $0^\\circ \\le \\theta \\le 180^\\circ,$ for which $\\csc \\theta = 2.$\n Solve this question step by step and put your final answer in \\boxed{}.",
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the positive difference between the radii of two circles if the circumference of a larger circle is $\\frac{2}{5}$ times the circumference of a smaller circle? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
... | math | {
"ground_truth": "\\frac{3}{5} r_s",
"style": "rule"
} | {
"answer": "\\frac{3}{5} r_s",
"difficulty": 1,
"has_image": true,
"id": 258405,
"index": 67,
"question": "What is the positive difference between the radii of two circles if the circumference of a larger circle is $\\frac{2}{5}$ times the circumference of a smaller circle? Express your answer in terms of ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nHow many cubic yards of stone are necessary to build the gate?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "9259",
"style": "rule"
} | {
"answer": "9259",
"difficulty": 1,
"has_image": true,
"id": 258533,
"index": 68,
"question": "A plane gate is 250 feet long, 50 feet wide, and 20 feet high. How many cubic yards of stone are necessary to build the gate?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split"... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the ratio of the original area to the enlarged area, and by what factor does the circumference increase?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{9}, 3",
"style": "rule"
} | {
"answer": "\\frac{1}{9}, 3",
"difficulty": 1,
"has_image": true,
"id": 258587,
"index": 69,
"question": "A circular park has its diameter increased by a factor of 3. Simultaneously, the circumference is noted to be thrice its original length as well. What is the ratio of the original area to the enlarged ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the length of the side that is common to both rectangles?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6",
"style": "rule"
} | {
"answer": "6",
"difficulty": 1,
"has_image": true,
"id": 258779,
"index": 70,
"question": "The ratio of the areas of two rectangles is 3:5, and the difference of their perimeters is 12 meters. What is the length of the side that is common to both rectangles?\n Solve this question step by step and put your... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the distance between the points.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{5490}",
"style": "rule"
} | {
"answer": "\\sqrt{5490}",
"difficulty": 1,
"has_image": true,
"id": 258876,
"index": 71,
"question": "Find the distance between the points $(-22,35)$ and $(17,-28)$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the volume of the cone, in cubic centimeters? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "108\\pi",
"style": "rule"
} | {
"answer": "108\\pi",
"difficulty": 1,
"has_image": true,
"id": 259410,
"index": 72,
"question": "What is the volume of the cone of radius $6$ cm and height $9$ cm, in cubic centimeters? Express your answer in terms of $\\pi$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nGraph the function \\( y = \\cos^2 x \\). Include key points and describe the transformation used to sketch the graph.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Graph described as above",
"style": "rule"
} | {
"answer": "Graph described as above",
"difficulty": 1,
"has_image": true,
"id": 259654,
"index": 73,
"question": "Graph the function \\( y = \\cos^2 x \\). Include key points and describe the transformation used to sketch the graph.\n Solve this question step by step and put your final answer in \\boxed{}... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIn the myriad of shapes and forms available, it was ayoucic triangle with its sixty-sixth wash day. What is the width of its base?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{1666}",
"style": "rule"
} | {
"answer": "\\sqrt{1666}",
"difficulty": 1,
"has_image": true,
"id": 260069,
"index": 74,
"question": "In the myriad of shapes and forms available, it was ayoucic triangle with its sixty-sixth wash day. This triangle is a right-angled isosceles triangle. It has an area of 833 square feet. What is the width... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the reflection of the point $(2, -3, 4)$ in the $xy$-plane?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(2, -3, -4)",
"style": "rule"
} | {
"answer": "(2, -3, -4)",
"difficulty": 1,
"has_image": true,
"id": 260275,
"index": 75,
"question": "What is the reflection of the point $(2, -3, 4)$ in the $xy$-plane?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\n12. \n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "354.26^\\circ",
"style": "rule"
} | {
"answer": "354.26^\\circ",
"difficulty": 1,
"has_image": true,
"id": 260321,
"index": 76,
"question": "12. \\(\\sin \\theta = -1/10\\) and \\(270^\\circ < \\theta < 360^\\circ\\)\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA cylindrical tank is filled with water. What is the radius of the tank's circular base?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "11.28",
"style": "rule"
} | {
"answer": "11.28",
"difficulty": 1,
"has_image": true,
"id": 260432,
"index": 77,
"question": "A cylindrical tank with a height of 10 meters is filled with water. The volume of water in the tank is 4000 cubic meters. What is the radius of the tank's circular base?\n Solve this question step by step and pu... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nExpress the statement \"Plane $\\alpha$ passes through line $l$\" using set notation: ______.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "l \\subset \\alpha",
"style": "rule"
} | {
"answer": "l \\subset \\alpha",
"difficulty": 1,
"has_image": true,
"id": 260586,
"index": 78,
"question": "Express the statement \"Plane $\\alpha$ passes through line $l$\" using set notation: ______.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subs... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the midpoint of the segment $AB$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(-2, -1)",
"style": "rule"
} | {
"answer": "(-2, -1)",
"difficulty": 1,
"has_image": true,
"id": 260788,
"index": 80,
"question": "Find the midpoint of the segment $AB$, if $A$ has coordinates $(-12, -14)$ and $B$ has coordinates $(8, 12)$.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nFind the center of the circle defined by the equation \\[(x-2)^2+ (y+5)^2=65.\\]\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(2, -5)",
"style": "rule"
} | {
"answer": "(2, -5)",
"difficulty": 1,
"has_image": true,
"id": 261341,
"index": 81,
"question": "Find the center of the circle defined by the equation \\[(x-2)^2+ (y+5)^2=65.\\]\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13000"
} | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA rectangular township is represented on the map. What is the area of the actual township? What is the area of the corresponding region on the map?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "150, 54",
"style": "rule"
} | {
"answer": "150, 54",
"difficulty": 1,
"has_image": true,
"id": 261583,
"index": 82,
"question": "On a map, 3 centimeters represents 5 miles. A rectangular township is represented on the map with dimensions 6 centimeters by 9 centimeters. What is the area of the actual township? What is the area of the cor... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nGiven a vector \\( \\mathbf{w} \\), express \\( \\mathbf{w} \\) in component form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\langle -2, 0 \\rangle",
"style": "rule"
} | {
"answer": "\\langle -2, 0 \\rangle",
"difficulty": 1,
"has_image": true,
"id": 261762,
"index": 83,
"question": "Given a vector \\( \\mathbf{w} \\) with direction angle \\( \\theta = 180^\\circ \\) and magnitude \\( \\|\\mathbf{w}\\| = 2 \\), express \\( \\mathbf{w} \\) in component form.\n Solve this que... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat are the dimensions of the rectangle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "14 \\text{ feet} \\times 6 \\text{ feet}",
"style": "rule"
} | {
"answer": "14 \\text{ feet} \\times 6 \\text{ feet}",
"difficulty": 1,
"has_image": true,
"id": 261989,
"index": 84,
"question": "What are the dimensions of the rectangle with width 6 feet and perimeter 40 feet?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIn what direction should you face? (Hint: 3600 degrees equals one full circle around the compass.)\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "west",
"style": "rule"
} | {
"answer": "west",
"difficulty": 1,
"has_image": true,
"id": 262025,
"index": 85,
"question": "In what direction should you face if you start facing east and you turn an angle of 900 degrees? (Hint: 3600 degrees equals one full circle around the compass.)\n Solve this question step by step and put your fin... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nExpress sin 255 degrees in terms of sines, cosines, and circles.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"style": "rule"
} | {
"answer": "-\\frac{\\sqrt{6} + \\sqrt{2}}{4}",
"difficulty": 1,
"has_image": true,
"id": 262080,
"index": 86,
"question": "Express sin 255 degrees in terms of sines, cosines, and circles.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nBen claims that he can calculate the length of the third side of a right triangle knowing only the lengths of the two other sides. What is the 3rd side? Express your answer in simplest radical form.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "... | math | {
"ground_truth": "\\sqrt{74}",
"style": "rule"
} | {
"answer": "\\sqrt{74}",
"difficulty": 1,
"has_image": true,
"id": 262228,
"index": 87,
"question": "Ben claims that he can calculate the length of the third side of a right triangle knowing only the lengths of the two other sides. What is the 3rd side given that the legs (the 2 non-hypotenuse sides) are 5... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nSketch the curve whose equation is \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\).\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Ellipse centered at the origin with vertices at (\\pm a, 0) and (0, \\pm b).",
"style": "rule"
} | {
"answer": "Ellipse centered at the origin with vertices at (\\pm a, 0) and (0, \\pm b).",
"difficulty": 1,
"has_image": true,
"id": 262257,
"index": 88,
"question": "Sketch the curve whose equation is \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\).\n Solve this question step by step and put your final answ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhich drum would require the greatest amount of metal to manufacture its surface area?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Second Drum",
"style": "rule"
} | {
"answer": "Second Drum",
"difficulty": 1,
"has_image": true,
"id": 262452,
"index": 89,
"question": "Three cylindrical drums each have a diameter of 10 inches. If the first drum's height is 5 inches, the second's height is 7 inches, and the third drum's height is 3 inches, which drum would require the gre... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the area of a triangle?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{n^2 + n}{2}",
"style": "rule"
} | {
"answer": "\\frac{n^2 + n}{2}",
"difficulty": 1,
"has_image": true,
"id": 262714,
"index": 90,
"question": "What is the area of a triangle if its base and height are consecutive integers?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001-13... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nPlease describe how you would draw a square pyramid given that each side of the square base measures 4 units?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "Draw a square with side length 4 units, locate the center of the square, place the apex directly above the center at a height of \\(\\frac{7}{4}\\) units, and connect the apex to each vertex of the square.",
"style": "rule"
} | {
"answer": "Draw a square with side length 4 units, locate the center of the square, place the apex directly above the center at a height of \\(\\frac{7}{4}\\) units, and connect the apex to each vertex of the square.",
"difficulty": 1,
"has_image": true,
"id": 262730,
"index": 91,
"question": "Please desc... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nSquare $ABCD$. What are the Cartesian coordinates of point $D$?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(0, 12)",
"style": "rule"
} | {
"answer": "(0, 12)",
"difficulty": 1,
"has_image": true,
"id": 262923,
"index": 92,
"question": "Square $ABCD$ has each side of length $12.$\tOne of the vertices of the square is the origin.\tWhat are the Cartesian coordinates of point $D$?\n Solve this question step by step and put your final answer in \... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nConvert the point in polar coordinates to rectangular coordinates.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(0, 6)",
"style": "rule"
} | {
"answer": "(0, 6)",
"difficulty": 1,
"has_image": true,
"id": 263665,
"index": 93,
"question": "Convert the point \\(\\left( 6, \\frac{5 \\pi}{2} \\right)\\) in polar coordinates to rectangular coordinates.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe equation of the perpendicular bisector of line segment AB is __________.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "x+2y-3=0",
"style": "rule"
} | {
"answer": "x+2y-3=0",
"difficulty": 1,
"has_image": true,
"id": 263825,
"index": 94,
"question": "Given two points A (-2, 0) and B (0, 4), the equation of the perpendicular bisector of line segment AB is __________.\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "t... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWrite the center and radius of the circle.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\left( \\frac{5}{2}, 2 \\right), \\frac{\\sqrt{41}}{2}",
"style": "rule"
} | {
"answer": "\\left( \\frac{5}{2}, 2 \\right), \\frac{\\sqrt{41}}{2}",
"difficulty": 1,
"has_image": true,
"id": 263909,
"index": 95,
"question": "Write the center and radius of the circle given by the equation {\\displaystyle x^2 + y^2 - 5x - 4y + 0 = 0}\n Solve this question step by step and put your fin... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nA square and a circle are cut from the same piece of fabric. What is the area of the remaining fabric after the shapes are cut out?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "3.43",
"style": "rule"
} | {
"answer": "3.43",
"difficulty": 1,
"has_image": true,
"id": 263971,
"index": 96,
"question": "A square and a circle are cut from the same piece of fabric. If the square has a 3-inch side length and the circle has a radius of 2 inches, and the total area of the fabric used is 25 square inches, what is the ... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nWhat is the area of the garden?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "25\\pi",
"style": "rule"
} | {
"answer": "25\\pi",
"difficulty": 1,
"has_image": true,
"id": 264242,
"index": 97,
"question": "A circular garden is surrounded by a fence. The circumference of the fence is $20\\pi$ feet, and the distance from the center of the garden to the fence is 5 feet. What is the area of the garden?\n Solve this q... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nIf a rectangular floor is to be covered with a square carpet, what is the total area of the floor that remains uncovered?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "64",
"style": "rule"
} | {
"answer": "64",
"difficulty": 1,
"has_image": true,
"id": 264269,
"index": 98,
"question": "If a rectangular floor with dimensions 10 m long and 8 m wide is to be covered with a square carpet with sides measuring 4 m, what is the total area of the floor that remains uncovered?\n Solve this question step b... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nHow long can the third side be?\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1 < x < 27",
"style": "rule"
} | {
"answer": "1 < x < 27",
"difficulty": 1,
"has_image": true,
"id": 264299,
"index": 99,
"question": "A triangle has two side lengths, 13 inches and 14 inches. How long can the third side be?\n Solve this question step by step and put your final answer in \\boxed{}.",
"split": "train",
"subset": "10001... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nThe points $A$ and $B$ are the endpoints of the diameter of a circle. Find the coordinates of the center of the circle.\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "(2, 3)",
"style": "rule"
} | {
"answer": "(2, 3)",
"difficulty": 1,
"has_image": true,
"id": 264522,
"index": 100,
"question": "In a Cartesian coordinate system, the points $A(0, 7)$ and $B(4, -1)$ are the endpoints of the diameter of a circle. Find the coordinates of the center of the circle.\n Solve this question step by step and put... | |
CMU-POPE/ODA_geometry_image_filtered:image | [
{
"content": "<image>\nVerify the trigonometric identity: \n$$\n\\sec^{4} \\theta - \\sec^{2} \\theta = \\tan^{4} \\theta + \\tan^{2} \\theta\n$$\n Solve this question step by step and put your final answer in \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "\\sec^{4} \\theta - \\sec^{2} \\theta = \\tan^{4} \\theta + \\tan^{2} \\theta",
"style": "rule"
} | {
"answer": "\\sec^{4} \\theta - \\sec^{2} \\theta = \\tan^{4} \\theta + \\tan^{2} \\theta",
"difficulty": 1,
"has_image": true,
"id": 264529,
"index": 101,
"question": "Verify the trigonometric identity: \n$$\n\\sec^{4} \\theta - \\sec^{2} \\theta = \\tan^{4} \\theta + \\tan^{2} \\theta\n$$\n Solve this q... |
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