query stringlengths 22 409 | positive_passages listlengths 1 1 | negative_passages listlengths 12 12 | score float64 56 100 |
|---|---|---|---|
Query: How can I calculate the area when I only know the lengths of two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 3446,
"text": "['', 'A right triangle has a hypotenuse of length 15 cm and one acute angle measuring 30 degrees. Calculate the area of the triangle using trigonometry. Round your answer to the nearest hundredth.\\nTo find the area of the triangle, we first need to find the lengths of the other t... | 96 |
Query: If I have a triangle and measure two sides and the angle between them, what formula can I use to find the area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: What is the formula for finding the area of a triangle if you only have the lengths of two sides and the included angle? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: Imagine I'm building a triangular garden bed and I've measured two sides and the angle between them, how do I figure out its area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: When the sides of a triangle are 5m and 7m, and the included angle is 60 degrees, how do I find the area? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 5690,
"text": "['', 'A triangle has two sides of length 10 units and 15 units, with a included angle of 60 degrees. What is the area of the triangle?\\nTo find the area of the triangle, we can use the formula for the area of a triangle when two sides and the included angle are given:\\n\\nArea =... | 96 |
Query: How can I calculate the area of a triangle if I only know two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 35251,
"text": "['', '\\\\section{Area of Triangle in Terms of Two Sides and Angle}\\nTags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles\\n\\n\\\\begin{theorem}\\nThe area of a triangle $ABC$ is given by:\\n:$\\\\displaystyle ... | 96 |
Query: I'm trying to figure out the formula for finding the area of a triangle when given two side lengths and the included angle. Can you help? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37184,
"text": "['', 'A triangle has two sides of length 5 cm and 7 cm, and the included angle between them is 60 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2)ab * sin(C)\\n\\nwhere a and b are the lengths... | 96 |
Query: If I have a triangular garden bed and want to plant flowers in it, how much space will I have if I know two sides and the angle between them? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 37085,
"text": "['', 'A triangle has two sides, which measure 12cm and 14 cm, and the angle between them is 67 degrees. Find the area of the triangle using trigonometry.\\nTo find the area of the triangle, we can use the formula:\\n\\nArea = (1/2) * a * b * sin(C)\\n\\nwhere a and b are the leng... | 96 |
Query: Could you explain why the area of a triangle is calculated as half the product of two sides times the sine of the included angle? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 35251,
"text": "['', '\\\\section{Area of Triangle in Terms of Two Sides and Angle}\\nTags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles\\n\\n\\\\begin{theorem}\\nThe area of a triangle $ABC$ is given by:\\n:$\\\\displaystyle ... | 96 |
Query: What if the angle between the two sides is obtuse? | [
{
"doc_id": 17476,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C... | [
{
"doc_id": 29121,
"text": "['', '\\\\section{Measurements of Common Angles/Obtuse Angle}\\nTags: Angles\\n\\n\\\\begin{theorem}\\nAn obtuse angle measures $\\\\theta$, where:\\n: $90^\\\\circ < \\\\theta < 180^\\\\circ$\\nor:\\n: $\\\\dfrac \\\\pi 2 < \\\\theta < \\\\pi$\\n\\\\end{theorem}\\n\\n\\\\begin{p... | 88 |
Query: How can I summarize a set of numbers with one value? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 66228,
"text": "['', 'from math import log2\\n \"\"\"You are given a **binary string** `s`, and a **2D** integer array `queries` where `queries[i] = [firsti, secondi]`.\\n\\nFor the `ith` query, find the **shortest substring** of `s` whose **decimal value**, `val`, yields `secondi` when **bit... | 96 |
Query: When averaging a group of temperatures, what formula do I use? | [
{
"doc_id": 343,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a se... | [
{
"doc_id": 3010,
"text": "['', \"There are 2 scales of temp A & B. It was given A 14 to 133, B 36 to 87. Find the temp when temp of A is equal to temp of B\\nAnswer Choices: (A) 52.5 (B) 53.5 (C) 51.5 (D) 54.5 (E) 55.5\\nLet's think about the multi-choice question.\\nActually, when corresponding ranges of ... | 96 |
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