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- ...ractal, its structure remains similar at every scale. This concept of self-similarity is a hallmark of fractals, making them distinct from more traditional geome ...xample of a fractal, this curve is constructed by repeatedly subdividing a triangle, resulting in a shape with infinite perimeter but finite area.2 KB (302 words) - 10:17, 27 September 2024
- ...gle ABC</math>. Hence, <math>\triangle ABC \sim \triangle ADP</math> by AA similarity and <math>\frac{AB}{AD}=\frac{BC}{DP}\implies DP=\frac{(AD)(BC)}{(AB)}.</ma ...gle CAD \implies \angle BAD=\angle CAP,</math> so <math>\triangle BAD\sim \triangle CAP.</math> This yields <math>\frac{AB}{AC}=\frac{BD}{CP}\implies CP=\frac{6 KB (922 words) - 16:34, 13 January 2025
- A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}4 KB (631 words) - 20:16, 8 October 2024
- But since this is a [[spiral similarity]], we also know that the triangles <math>ABD, \; PBC </math> are also simil Now, by the [[triangle inequality]], we have <math>AP + PC \ge AC </math>. Multiplying both sides3 KB (602 words) - 08:01, 7 June 2023
- ...thety often means that we can use properties of similarity. Additionally, similarity (especially with circles) where parallel lines are used can indicate that h ...ath> <math>\omega_{B},</math> and <math>\omega_{C}</math>. If the sides of triangle <math>ABC</math> are <math>13,</math> <math>14,</math> and <math>15,</math>3 KB (533 words) - 12:51, 2 September 2024
- ...math> passes through the vertices <math>A</math> and <math>C</math> of the triangle <math>ABC</math> and intersects the segments <math>AB</math> and <math>BC</ ...> and <math>M_2</math> be the midpoint of <math>NC</math>. Thus the spiral similarity must also send <math>M_1</math> to <math>M_2</math> and so <math>BMM_1 M_2<3 KB (496 words) - 12:35, 18 January 2023
- A '''right triangle''' is any [[triangle]] with an angle of 90 degrees (that is, a [[right angle]]). ...is labeled <math>c</math>. The other two sides are called the legs of the triangle.3 KB (499 words) - 22:41, 11 June 2022
- ...s the center of the square. Label the new triangle that we created <math>\triangle OGJ</math>. Now we know that rotation preserves angles and side lengths, s ...g \triangle FOG,</math> so <math>FG=400</math>. Now we have a right <math>\triangle BFG</math> with legs <math>x</math> and <math>500-x</math> and hypotenuse <13 KB (2,080 words) - 12:14, 23 July 2024
- ...r</math> to one of the sides creating an <math>r,r, r\sqrt{2}</math> right triangle. This means that we have <math>r + r\sqrt{2} = 8\sqrt{2}</math> so <math>r ...rom <math>O</math> to <math>AB</math>, which forms a <math>45-45-90</math> triangle. The length of the perpendicular is <math>\frac{r}{\sqrt{2}}</math>. Note a4 KB (707 words) - 10:11, 16 September 2021
- We proceed using complex numbers. Triangle <math>F_2OF_1'</math> is isosceles with side lengths <math>13,13,20</math>. ...ction as in Solution 2. As <math>OF_1'=OF_2=13</math>, we know that <math>\triangle OF_1'F_2</math> is isosceles. Hence <math>\angle F_2F_1'O=\angle F_1'F_2O</12 KB (2,001 words) - 19:26, 23 July 2024
- In [[triangle]] <math> ABC, AB=13, BC=15, </math> and <math>CA = 14. </math> Point <math> ...= 14-SC = \frac{52}{5}</math>. We can also use the Pythagorean theorem on triangle <math>AQB</math> to determine that <math>BQ = \frac{33}{5}</math>.14 KB (2,340 words) - 15:38, 21 August 2024
- ...[[ratio]] between the area of triangle <math> ABC </math> and the area of triangle <math> EBD </math> is <math> m/n, </math> where <math> m </math> and <math> ...as <math>AC \parallel DE</math>, it follows that <math>\triangle ABC \sim \triangle EFD</math>.3 KB (486 words) - 21:15, 7 April 2023
- ...are given that <math>AE=8</math> and <math>BE=17</math>, the length of the triangle is given and the height depends solely on the length of <math>CF</math>. Le ...hat <math>FF'</math> is perpendicular to <math>EB</math>. This makes right triangle <math>FF'E</math>, Also, note that <math>F'B</math> has length of <math>3</9 KB (1,500 words) - 19:06, 8 October 2024
- ...<math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>. ...onding side on the large triangle is <math>12x</math>, and the area of the triangle is <math>12^2 = \boxed{144}</math>.4 KB (726 words) - 12:39, 13 August 2023
- ...</math> and <math>C</math> divide the square into three parts, two [[right triangle]]s and a [[parallelogram]]. Using the smaller side of the parallelogram, < ...EL</math> is also similar to <math>\triangle CDF</math> by <math>AA</math> similarity. Thus, <math>\frac{FC}{EC} = \frac{DC}{LC}</math>, and the expression simpl3 KB (484 words) - 20:40, 2 March 2020
- ...ent]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>d</math>, find <mat ...milar]] (<math>\triangle ABC \sim \triangle DPD' \sim \triangle PEE' \sim \triangle F'PF</math>). The remaining three sections are [[parallelogram]]s.11 KB (1,879 words) - 20:04, 8 December 2024
- ...<math>\triangle BPQ \cong \triangle DRS</math>, <math>\triangle APS \cong \triangle CRQ</math>). Quickly we realize that <math>O</math> is also the center of t ...<math>\triangle ABD \sim \triangle OQP</math> by <math>AA</math> [[similar triangle|similar]]ity. From here, it's clear that8 KB (1,270 words) - 22:36, 27 August 2023
- ...onding angles, and thus by angle-angle similarity, <math>\triangle QOR\sim\triangle SDB</math>.11 KB (1,811 words) - 19:23, 29 December 2024
- ...=90^\circ.\,</math> Then the area of the set of all fold points of <math>\triangle ABC\,</math> can be written in the form <math>q\pi-r\sqrt{s},\,</math> wher ..., O_{BC}, O_{CA}</math> are, respectively, the [[circumcenter]]s of <math>\triangle PAB, PBC, PCA</math>. According to the problem statement, the circumcenters4 KB (717 words) - 21:20, 3 June 2021
- ...ABFG</math> is an isosceles trapezoid and <math>CDE</math> is an isosceles triangle, we have symmetry about the <math>xz</math>-plane. ...m <math>G</math> to <math>ABCD</math> has height <math>12/2 = 6</math>. By similarity, the x-coordinate of <math>G</math> is <math>-6/2 = -3</math>. Then <math>G7 KB (1,181 words) - 14:56, 1 July 2024