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- ...[triangles]] are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties. In a quadrilateral <math>ABCD</math>:1 KB (162 words) - 20:39, 9 March 2024
- If <math>\angle \text{CBD}</math> is a [[right angle]], then this protractor indicates that the measure of <math>\ pair A,B,C,D;1 KB (160 words) - 16:53, 17 December 2020
- ...point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad2 KB (299 words) - 15:29, 5 July 2022
- In triangle <math>ABC,\,</math> angle <math>C</math> is a right angle and the altitude from <math>C\,</math> meets <math>\overline{AB Since <math>\triangle ABC \sim \triangle CBD</math>, we have <math>\frac{BC}{AB} = \frac{29^3}{BC} \Longrightarrow BC^23 KB (534 words) - 16:23, 26 August 2018
- ...name of a point represent the mass located there. Since we are looking for a ratio, we assume that <math>AB=120</math>, <math>BC=169</math>, and <math>C First, reflect point <math>F</math> over angle bisector <math>BD</math> to a point <math>F'</math>.8 KB (1,382 words) - 00:37, 12 July 2024
- ...ath>AC \perp BD</math>, it follows that <math>\triangle BAC \sim \triangle CBD</math>, so <math>\frac{x}{\sqrt{11}} = \frac{y}{x} \Longrightarrow x^2 = y\ ...th>\overline{CD}</math>. Then <math>AE = x</math>, and <math>ADE</math> is a [[right triangle]]. By the [[Pythagorean Theorem]],4 KB (584 words) - 19:35, 7 December 2019
- ...lic, since <math>\angle AEB=\angle AFB</math>. We then have, from Power of a Point, that <math>CE\cdot CA=CF\cdot CB</math>. In other words, <math>1\cdo ...{DB}{AB}\cdot [ABC]=\frac{x}{9}</math> and <math>[CFD]=\frac{CF}{CB}\cdot [CBD]=\frac{2}{3}\cdot \frac{x}{9}=\frac{2x}{27}</math>. The desired ratio is th3 KB (518 words) - 16:54, 25 November 2015
- An '''angle''' is the [[union]] of two [[ray]]s with a common [[endpoint]]. The common endpoint of the rays is called the [[verte ...most common form is <math>\angle ABC</math>, read "angle ABC", where <math>A,C</math> are points on the sides of the angle and <math>B</math> is the ver4 KB (597 words) - 18:39, 9 May 2024
- ...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...of the circumcircle. But <math>AZ = YP</math> (since <math>AZYP</math> is a rectangle).2 KB (417 words) - 17:24, 21 July 2018
- Let <math>ABCD </math> be a convex quadrilateral with <math>AB=BC=CD </math>, <math>AC \neq BD </math>, ...ath>, <math> \angle BAC = \angle ACB </math>, and similarly, <math> \angle CBD = \angle BDC </math>. Since <math> \angle CEB = \angle AED </math>, by con3 KB (566 words) - 23:59, 14 September 2014
- <math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\lef pair C=(0,0),A=(0,3),B=(4,0),D=(4-2.28571,1.71429);6 KB (951 words) - 16:31, 2 August 2019
- <math>\mathrm{(A)}\ 75\qquad\mathrm{(B)}\ 80\qquad\mathrm{(C)}\ 85\qquad\mathrm{(D)}\ 90\qqu ...solution requires the use of cyclic quadrilateral properties but could be a bit time-consuming during the contest.12 KB (1,944 words) - 17:15, 20 January 2024
- ...milarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the com Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same o28 KB (4,863 words) - 00:29, 16 December 2023
- Triangle <math>ABC</math> has a right angle at <math>B</math>. Point <math>D</math> is the foot of the alti pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));5 KB (879 words) - 18:57, 30 April 2024
- The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of <math>\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)14 KB (1,872 words) - 15:23, 17 January 2023
- ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.4 KB (718 words) - 18:16, 17 September 2012
- ...> is cyclic if and only if <math>\overline{BG}</math> bisects <math>\angle CBD</math>. pair A = (-.6, .8), B = (.6, .8), C = (.9, -sqrt(.19)), D = (-.9, -sqrt(.19)), G =6 KB (973 words) - 19:24, 18 October 2018
- <math>\text{(A)}\ 14 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 14\sqr pair A, B, C, D, M;6 KB (899 words) - 01:41, 5 July 2023
- <math>\text{(A) } \dfrac{9}{8} \qquad \text{(B) } \dfrac{5}{3} \qquad \text{(C) } 2 \qquad pair A, B, C, D, E, I;3 KB (508 words) - 20:02, 6 August 2024
- Let <math>ABCD</math> be a cyclic quadrilateral. The side lengths of <math>ABCD</math> are distinct in <math>\textbf{(A)}\ \sqrt{\dfrac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)2 KB (262 words) - 16:43, 15 February 2021