GEOMETRY FUNDAMENTALS - UNIT 11 Essays (556 words) - Geometry

GEOMETRY FUNDAMENTALS - UNIT 11 REVIEW SIMILAR POLYGONS Cosine ratioIn a right triangle, the side adjacent to an acute angle over the hypotenuse.Geometric meanFor any positive real numbers a, b, and x if then x is called the geometric mean between a and b.Projection of a point on a lineThe point where a perpendicular through the point to the line intersects the line.Projection of a segment on a lineThe portion of a line with endpoints that are the projections of the endpoints of the segment.ProportionAn equation that states that two ratios are equal.RatioThe comparison of two numbers by division. The quotient is the ratio of the two numbers.Similar polygonsPolygons whose vertices can be matched in a one-to-one correspondence so that corresponding angles are equal and corresponding sides are in proportion.Sine ratioIn a right triangle, the side opposite an acute angle over the hypotenuse.Tangent ratioIn a right triangle, the side opposite an acute angle over the side adjacent to the acute angle. P-15:If the three angles of one triangle are equal to the three angles of another triangle, then the triangles are similar. (AAA) Cross Product property If , then ad = bc . Equivalent Forms property . The cross product of each proportion gives the same equation, ad = bc . Denominator Sum property Denominator Difference property Numerator-Denominator Sum property 5-1:If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.5-2:If two sides of one triangle are proportional to two sides of another and the included angles are equal, then the triangles are similar.5-3:If three sides of one triangle are proportional to three sides of another, then the triangles are similar.5-4:Similarity of polygons is reflexive, symmetric, and transitive.5-5:If two polygons are similar, then the ratio of their perimeters is the same as the ratio of any pair of corresponding sides.5-6:If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally.5-7:If a ray bisects an angle of a triangle, then it divides the opposite side into segments with lengths proportional to the lengths of the other two sides of the triangle.5-8:If two triangles are similar, then the lengths of corresponding altitudes have the same ratio as the lengths of any pair of correspon ding sides.5-9:If the altitude to the hypotenuse of a right triangle is drawn, then the two triangles formed are similar to each other and similar to the given triangle.5-10:In a right triangle the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse. (Pythagorean Theorem)5-11:If the sum of the squares of two sides of a triangle equals the square of the third side, then the triangle is a right triangle.5-12:In a 30-60-90 triangle, the hypotenuse is twice the short leg, and the longer leg is times the short leg5-13:If each acute angle of a triangle is 45, then the measure of the hypotenuse is times the leg.

Conjugation of the Spanish Verb Saber

Conjugation of the Spanish Verb Saber Saber, a common Spanish verb usually meaning to know in the sense of having knowledge, is highly irregular. Both the stem and the endings can take unexpected forms. Saber should not be confused with conocer, which also means to know, but in the sense of being familiar with a person. Conocer also is conjugated irregularly Irregular forms are shown below in boldface. Pronouns are shown for convenience and clarity; they often are omitted in everyday speech and writing. Infinitive of Saber saber (to know) Gerund of Saber sabiendo (knowing) Participle of Saber sabido (known) Present indicative of Saber sà © (I know), tà º sabes (you know), usted/à ©l/ella sabe (you know, he/she knows), nosotros/as sabemos (we know), vosotros/as sabà ©is (you know), ustedes/ellos/ellas saben (you/they know) Preterite Indicative of Saber yo supe (I knew), tà º supiste (you knew), usted/à ©l/ella supo (you, he/she knew), nosotros/as supimos (we knew), vosotros/as supisteis (you knew), ustedes/ellos/ellas supieron (they knew) Imperfect Indicative of Saber yo sabà ­a (I knew), tà º sabà ­as (you knew), usted/à ©l/ella sabà ­a (you/he/she knew), nosotros/as sabà ­amos (we knew), vosotros/as sabà ­ais (you knew), ustedes/ellos/ellas sabà ­an (you/they knew) Future Indicative of Saber yo sabrà © (I will know), tà º sabrs (you will know), usted/à ©l/ella sabr (you/he/she will know), nosotros/as sabremos (we will know), vosotros/as sabrà ©is (you will know), ustedes/ellos/ellas sabrn (they will know) Conditional of Saber yo sabrà ­a (I would know), tà º sabrà ­as (you would know), usted/à ©l/ella sabrà ­a (you/he/she would know), nosotros/as sabrà ­amos (we would know), vosotros/as sabrà ­ais (you would know), ustedes/ellos/ellas sabrà ­an (you/they would know) Present Subjunctive of Saber que yo sepa (that I know), que tà º sepas (that you know), que usted/à ©l/ella sepa (that you/he/she know), que nosotros/as sepamos (that we know), que vosotros/as sepis (that you know), que ustedes/ellos/ellas sepan (that you/they know) Imperfect Subjunctive of Saber que yo supiera/supiese (that I knew), que tà º supieras/supieses (that you knew), que usted/à ©l/ella supiera/supiese (that you/he/she knew), que nosotros/as supià ©ramos/supià ©semos (that we knew), que vosotros/as supierais/supieseis (that you knew), que ustedes/ellos/ellas supieran/supiesen (that you/they knew) Imperative of Saber sabe tà º (know), no sepas tà º (dont know), sepa usted (know), sepamos nosotros/as (let us know), sabed vosotros/as (know), no sepis vosotros/as (dont know), sepan ustedes (know) Present Perfect Indicative of Saber yo he sabido, tà º has sabido, usted/à ©l/ella ha sabido, nosotros/as hemos sabido, vosotros habà ©is sabido, ustedes/ellos/ellas han sabido (I have known, you have known, she has known, etc.) Pluperfect (Past Perfect Indicative of Saber) yo habà ­a sabido, tà º habà ­a sabido, usted/à ©l/ella habà ­a sabido, nosotros/as habà ­amos sabido, vosotros habà ­ais sabido, ustedes/ellos/ellas habà ­an sabido (that I had known, that you had known, that he had known, etc.) Future Perfect Indicative of Saber yo habrà © sabido, tà º habrs sabido, usted/à ©l/ella habr sabido, nosotros/as habremos sabido, vosotros habrà ©is sabido, ustedes/ellos/ellas habrn sabido (I will have known, you will have known, she will have known, etc.) Present Perfect Subjunctive of Saber yo haya sabido, tà º hayas sabido, usted/à ©l/ella haya sabido, nosotros/as hayamos sabido, vosotros hayis sabido, ustedes/ellos/ellas hayan sabido (that I have known, that you have known, that he has known, etc.) Past Perfect Subjunctive of Saber yo hubiera/hubiese sabido, tà º hubieras/hubieses sabido, usted/à ©l/ella hubiera/hubieses sabido, nosotros/as hubià ©ramos/hubià ©semos sabido, vosotros hubierais/hubieseis sabido, ustedes/ellos/ellas hubieran/hubiesen sabido (that I had known, that you had known, that she had known, etc.) Conditional Perfect of Saber yo habrà ­a sabido, tà º habrà ­as sabido, usted/à ©l/ella habrà ­a sabido, nosotros/as habrà ­amos sabido, vosotros habrà ­ais sabido, ustedes/ellos/ellas habrà ­an sabido (I would have known, you would have known, he would have known, etc.) Progressive Tenses of Saber The many  progressive  tenses use the appropriate form  estar  followed by the  gerund,  sabiendo. It is not common, however, to use saber in a progressive form. Sample Sentences Using Forms of Saber Quiero saber lo que piensas. (I want to know what youre thinking. Infinitive.) No sà © que hacer. (I dont know what to do. Present tense.)  ¿Cuntos aà ±os tenà ­as cuando supiste la siguiente informacià ³n? (How old were you when you learned the following information? Note that in the preterite tense, which is used here, conocer often means to learn or to come to know.) Estbamos sorprendidos porque no sabà ­amos lo que habà ­a pasado. (We were surprised because we didnt know what had happened.) Hemos sabido por mucho tiempo que el ser humano posee el poder transformar la Tierra en formas a la vez terribles y majestuosas. (We have known for a long time that the human being has the power to transform the Earth in ways both terrible and majestic at the same time. Present perfect.) Sin muchos consejos, no sabrà ­amos lo que debemos estudiar. (Without a lot of advice, we wouldnt know what we should study.) Cuando veas las fotos, sabrs la verdad. (When you see the photos, you will know the truth. Future.) No me importa que Roxana sepa que soy su benefactor. (It doesnt matter that Roxana knows Im her benefactor. Present subjunctive.) Podrà ­a estar roto por semanas y no lo habrà ­an sabido. (It could be broken for weeks and they would not have known. Conditional perfect.)