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- A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. === Coordinate Definition ===9 KB (1,585 words) - 12:46, 2 September 2024
- ...cular. Drawing all four semi-axes divides the ellipse into 4 [[congruent (geometry)|congruent]] quarters. ...ystem]] with center <math>C = (h, k)</math> whose axes are parallel to the coordinate axes, with the horizontal semi-axis of length <math>a</math> and the vertic5 KB (892 words) - 20:52, 1 May 2021
- ...gth of the equilateral is <math>3\sqrt{3}</math>. We now use 3D coordinate geometry. [[Category:Intermediate Geometry Problems]]6 KB (980 words) - 20:45, 31 March 2020
- ...[circle]]s of [[diameter]] 1 are packed in the first [[quadrant]] of the [[coordinate plane]] as shown. Let region <math> \mathcal{R} </math> be the union of the The line passing through the [[tangent (geometry)|tangency point]] of the bottom left circle and the one to its right and th4 KB (731 words) - 16:59, 4 January 2022
- Note that the apex of the angle is not on the parallel lines. Set up a [[coordinate proof]]. [[Category:Intermediate Geometry Problems]]4 KB (709 words) - 00:50, 10 January 2022
- One possibility is to use the [[coordinate plane]], setting <math>B</math> at the origin. Point <math>A</math> will be [[Category:Introductory Geometry Problems]]6 KB (958 words) - 22:29, 28 September 2023
- [[Circle]]s with [[center_(geometry) | center]]s <math>(2,4)</math> and <math>(14,9)</math> have [[radius | rad ...n of <math>L_1</math> is <math>y=\frac{5}{12}x+\frac{19}{6}</math>, so the coordinate of this point is <math>\left(-\frac{38}{5},0\right)</math>. Hence the equat2 KB (295 words) - 01:24, 30 October 2024
- Let <math>S</math> be the [[set]] of all [[point]]s with [[coordinate]]s <math>(x,y,z)</math>, where <math>x</math>, <math>y</math>, and <math>z< [[Category:Intermediate Geometry Problems]]4 KB (495 words) - 00:36, 26 May 2024
- Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, ...>2</math> dimensions, place <math>4</math> circles in each quadrant in the coordinate plane. To find the circle enclosing these <math>4</math> circles, notice th2 KB (364 words) - 03:54, 16 January 2023
- Using the equation of the median from above, we can write the [[coordinate]]s of <math>A</math> as <math>(p,\ -5p + 107)</math>. The equation of <math [[Category:Intermediate Geometry Problems]]5 KB (852 words) - 20:23, 4 October 2023
- ...y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </math> and internally tangent to <math> w_1. </ma ...quation of the ellipse as laid out in previous solutions. We now scale the coordinate plane in the <math>x</math> direction by a factor of <math>\frac{\sqrt{3}}{12 KB (2,001 words) - 19:26, 23 July 2024
- ...oint of intersection of <math>EF, BB'</math> be <math>G</math>. Then the y-coordinate of <math>G</math> is <math>\frac{25}{2}</math>, so ===Solution 3 (Coordinate Bashing)===9 KB (1,500 words) - 19:06, 8 October 2024
- ===Solution 6 (Only simple geometry and algebra needed) === ===Solution 8 (Coordinate Bash)===14 KB (2,351 words) - 20:06, 8 December 2024
- === Solution 4 (coordinate geometry) === ...dinate of <math>D</math> is <math>5</math>, we see that the <math>y</math>-coordinate cannot be lower than <math>3</math>, that is, <math>\alpha \in \left[\sin^{20 KB (3,497 words) - 14:37, 27 May 2024
- We can redefine the coordinate system so that the center of the center circle is the origin, for easier ca * [[AIME Problems and Solutions]]6 KB (1,022 words) - 18:29, 22 January 2024
- To find <math>C</math>, we can again let the <math>x</math>-coordinate be <math>\frac{3}{2}</math> for simplicity. Note that <math>C</math> is <ma ...> This is because we know the projection will have the same <math>x</math>-coordinate as <math>D</math> and <math>C</math>, as both are <math>\frac{3}{2}.</math>6 KB (947 words) - 19:44, 26 November 2021
- ...= 25</math>. This forms an [[isosceles triangle]] <math>PQS</math>. The [[coordinate]]s of <math>S</math>, using the slope of <math>PR</math> (which is <math>-4 ...the x-coordinate is moving to the left <math>4+3=7</math> units and the y-coordinate is moving up <math>-3+4=1</math> units. The line through <math>(-8,5)</math8 KB (1,319 words) - 10:34, 22 November 2023
- First, consider the [[triangle]] in a [[coordinate system]] with [[vertex|vertices]] at <math>(0,0)</math>, <math>(9,0)</math> [[Category:Intermediate Geometry Problems]]4 KB (703 words) - 22:13, 30 August 2024
- are drawn in the coordinate plane for <math>k=-10,-9,-8,\ldots,9,10.\,</math> These 63 lines cut part [[Category:Intermediate Combinatorics Problems]]4 KB (721 words) - 15:14, 8 March 2021
- Without loss of generality, place the pyramid in a 3-dimensional coordinate system such that <math>A = (1,0,0),</math> <math>B = (0,1,0),</math> <math> [[Category:Intermediate Geometry Problems]]8 KB (1,172 words) - 20:57, 22 September 2022