ALL GEOMETRY FORMULAS COMPILED FOR SAT,GRE etc exams.

FORMULAS OF GEOMETRY FOR SAT,GRE,and Other COMPETITIVE EXAMS Page 1 STRAIGHT LINE ¾ A straight line is the shortest distance between two points. ¾ A straight line covers 180 . o ¾ If two straight lines intersect , then the opposite angles are equal and the adjacent angles add up to 180 . o In the above illustration, angle x is equal to angle y. We do not know enough about the diagram to make a determination whether angles r, s, t, and q are equal to each other or equal to any value at all. If the lines AB and ED are parallel to each other, then t = s and q = r o o o o PARALLEL LINES and TRANSVERSAL Lines l and m are parallel to each other, and line k is a transversal. Then the angles a and d are equal, and angles b and c are equal. It is also true that a = d o o = w = z . Likewise, it is o o true that b = c = x = y . o o o o EXAMPLE OF A PROBLEM TESTING YOUR UNDERSANDING OF TWO INTERSECTING STRAIGHT LINES We notice that 2Y = 3X + 20 and 2Y + 2X + 10 = 180 and 2X+10+3X + 20 = 180 We notice that if we combined the two angles 2 X+10 and 3X+20, we will set up an equation involving one variable X. We get: 2X+10+3X+20 = 180 5X + 30 = 180 or 5X = 150 or X = 30 o Now we can use the relationship 2Y = 3X+20 and solve for Y. We get: 2Y = 3(30) + 20 = 110 Or Y = 110/2 = 55 o  © MLICETS , an Educational Training Services Division of MLI Consulting, Inc . , New York U.S.A For use by participants registered in our GMAT, GRE, and SAT prep courses. Not to be reproduced or distributed without our written consent. Page 2 POLYGON ¾ A polygon is a closed figure formed by using three or more straight lines. Triangle is the simplest of polygons. ¾ The sum of the internal angles of a polygon of N sides is (N-2) € 180 o . For example, a triangle has three sides and the sum of the internal angles is (3-2) times 180 , or o 180 o . TRIANGLE ¾ A triangle is a polygon formed by 3 straight lines. ¾ The SUM Of the internal angles is 180 . o ¾ The side opposite the largest angle is the longest side of the triangle. ¾ Any side of a triangle must measure MORE than the positive difference of the other two sides, and LESS than the sum of the other two sides. If two sides are 3 and 7, then the third side must be longer than 4 and shorter than 10. TRIANGLE CONTINUED ¾ The area of a triangle is ¾ ½ € (Base) € (Height) ¾ The PERIMETER of a triangle is the sum of its three sides. ¾ The HEIGHT is the height of the vertical line drawn from the VERTEX of the triangle to its base, no matter what shape the triangle is. Vertex vertex H H SPECIAL TRIANGLES ISOSCELES TRIANGLE ¾ An isosceles triangle has two sides of equal length enclosing two equal angles with the third side. The two congruent angles could be any value, and we cannot determine what they are on the basis of a statement that the triangle is an isosceles triangle. ¾ The vertical line drawn from the vertex where the two line segments of equal length meet to the base will bisect the b a se in two equal halves. C Y o y o A D B AC = CB, AND /A = /B AND AD = DB ¾ The area of an isosceles Triangle is computed by using the formula: ½ € Base € Height = ½ € AB € CD SIMILAR TRIANGLES A triangle that is ‘SIMILAR’ to another is a SCALED or proportionately reduced version of the other. The lengths of the corresponding sides of the two triangles are in the same ratio, and the measures of their corresponding angles are equal.  © MLICETS , an Educational Training Services Division of MLI Consulting, Inc . , New York U.S.A For use by participants registered in our GMAT, GRE, and SAT prep courses. Not to be reproduced or distributed without our written consent. Page 3 SPECIAL TRIANGLES CONTINUED EQUILATERAL TRIANGLE ¾ An equilateral triangle is one in which ALL SIDES HAVE THE SAME LENGTH, and ALL INTERNAL ANGLES ARE 60 EACH. O ¾ The area of an equilateral triangle can be computed using the standard formula for the area of a triangle or by using the special formula that applies only to an equilateral triangle: ( _ 3) ( ¼ ) € side 2 ¾ THE HEIGHT OF AN EQUILATERAL TRIANGLE IS ( _ 3 )( ½ )side. ¾ We can compute the area of an equilateral triangle if we know the measure of the side or of the height, and do not require both. The area of an equilateral triangle in terms of its height is ( _ 3)( 1 / )height . 3 2 EXTERNAL ANGLE OF A TRIANGLE We can see that Z and 50 are supplementary angles adding up to 180 degrees. We can also see, using the properties of intersecting lines, that y is equal to 50 degrees. Z is the EXTERNAL ANGLE of the triangle, and is equal to the SUM OF THE TWO REMOTE INTERNAL ANGLES X and 60. Therefore, z = x + 60 RULE: ANY EXTERNAL ANGLE OF A TRIANGLE IS EQUAL TO THE SUM OF THE TWO REMOTE INTERNAL ANGLES. (Remote angles are the ones excluding the adjacent angle, in the above illustration 50 o ). EXAMPLE OF A PROBLEM TESTING YOUR UNDERSTANDING OF EXTERNAL ANGLE OF A TRIANGLE. In the figure above, Angle ACD is how many degrees greater than Angle BAC? We can see that Angle ABD is 50 , given o that angle BAD is 40 . Also, using our o knowledge of external angles, we can see that Angle ACD = Angle BAC + Angle ABC Angle ACD = Angle BAC + 50 o Therefore, Angle ACD - Angle BAC = 50 . o We can, therefore, conclude that Angle ACD is 50 degrees greater than Angle BAC.