Tiêu đề Math-Readings-3.pdf ✅

Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math III - 1 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! GEOMETRY POINTERS Euclidean geometry axioms As we have noted above, there is a set of the axioms – properties, that are considered in geometry as main ones and are adopted without a proof . Now, after introducing some initial notions and definitions we can consider the following sufficient set of the axioms, usually used in plane geometry. Axiom of belonging. Through any two points in a plane it is possible to draw a straight line, and besides only one. Axiom of ordering. Among any three points placed in a straight line, there is no more than one point placed between the two others. Axiom of congruence ( equality ) of segments and angles. If two segments (angles) are congruent to the third one, then they are congruent to each other. Axiom of parallel straight lines. Through any point placed outside of a straight line it is possible to draw another straight line, parallel to the given line, and besides only one. Axiom of continuity ( Archimedean axiom ). Let AB and CD be two some segments; then there is a finite set of such points A1 , A2 , ... , An , placed in the straight line AB, that segments AA1 , A1A2 , ... , An - 1An are congruent to segment CD, and point B is placed between A and An . We emphasize, that replacing one of these axioms by another, turns this axiom into a theorem, requiring a proof. So, instead of the axiom of parallel straight lines we can use as an axiom the property of triangle angles (“the sum of triangle angles is equal to 180 deg”). But then we should to prove the property of parallel lines. Straight line A general equation of straight line: Ах + Ву + С = 0 , where А and В aren't equal to zero simultaneously.
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math III - 2 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! Coefficients А and В are coordinates of normal vector of the straight line ( i.e. vector, perpendicular to the straight line ). At А = 0 straight line is parallel to the axis ОХ , at В = 0 straight line is parallel to the axis ОY . At В 0 we receive an equation of straight line with a slope: An equation of the straight line, going through the point ( х0 , у 0 ) and not parallel to the axis OY : у – у 0 = m ( x – х0 ) , where m is a slope, equal to tangent of an angle between the straight line and the positive direction of the axis ОХ . At А 0, В 0 and С 0 we receive an equation of straight line in segments on axes: where a = – C / A, b = – C / B. This line goes through the points ( a, 0 ) and ( 0, b ), i.e. it cuts off segments a and b long on the coordinate axes. An equation of straight line going through two different points ( х1, у 1 ) and ( х2, у 2 ):
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math III - 3 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! Parallel straight lines Two straight lines AB and CD ( Fig.11 ) are called parallel straight lines, if they lie in the same plane and don’t intersect however long they may be continued. The designation: AB|| CD. All points of one line are equidistant from another line. All straight lines, parallel to one straight line are parallel between themselves. It’s adopted that an angle between parallel straight lines is equal to zero. An angle between two parallel rays is equal to zero, if their directions are the same and 180 deg, if the directions are opposite. All perpendiculars (AB, CD, EF, and Fig.12) to the one straight line KM are parallel between themselves. Inversely, the straight line KM, which is perpendicular to one of parallel straight lines, is perpendicular to all others. A length of perpendicular segment, concluded between two parallel straight lines, is a distance between them. At intersecting two parallel straight lines by the third line, eight angles are formed (Fig.13), which are called two-by-two:
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math III - 4 - Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! 1) corresponding angles (1 and 5; 2 and 6; 3 and 7; 4 and 8 ); these angles are equal two-by-two: ( 1 = 5; 2 = 6; 3 = 7; 4 = 8 ); 2) alternate interior angles ( 4 and 5; 3 and 6 ); they are equal two-by-two; 3) alternate exterior angles ( 1 and 8; 2 and 7 ); they are equal two-by-two; 4) one-sided interior angles (3 and 5; 4 and 6 ); a sum of them two-by-two is equal to180 deg ( 3 + 5 = 180 deg; 4 + 6 = 180 deg); 5) one-sided exterior angles ( 1 and 7; 2 and 8 ); a sum of them two-by-two is equal to180 deg ( 1 + 7 = 180 deg; 2 + 8 = 180 deg). Angles with correspondingly parallel sides either are equal one to another, ( if both of them are acute or both are obtuse, 1 = 2, Fig.14 ), or sum of them is 180 deg ( 3 + 4 = 180 deg, Fig.15 ). Angles with correspondingly perpendicular sides are also either equal one to another ( if both of them are acute or both are obtuse ), or sum of them is 180 deg.