# gmat geometry

 area of a triangle (base x height)/2 area of a trapezoid area of any parallelogram  base x height area of a rhombus surface area the sum of all the facesto determine the SA of a rectangular solid, you must find the area of each faceto determine the SA of a cube, you only need the length of one side volume length x width x height parallelogram opposite sides and opposite angles are equal trapezoid one pair of opposite sides is parallel.   rhombus all sides are equal.  opposite angles are equal. sum of interior angles of a polygon (n-2) x 180 where n = the number of sides isosceles triangle if two sides are equal, their opposite angles are also equal45-45-90leg:leg:hypotenuse 1:1: x:x:x triangle inequality law the sum of any two sides of a triangle must be great than the third sidethe difference of any two sides must be less than  the third side pythagorean theorem a2 + b2 = c2 common right triangles 3-4-5; 32 + 42 = 52 (9 + 16 = 25); key multipliers: 6-8-10, 9-12-15, 12-16-205-12-13; 52 + 122 = 132 (25 + 144 = 169); key multipliers: 10-24-268-15-17; 82 + 152 = 172 (64 + 225 = 289) equilateral triangle all three sides (and all three angles) are equaleach angle is 60 degreestwo 30-60-90 triangle form an equilateral triangle 30-60-90 short leg : long leg : hypotenuse1 : : 2x : x : 2x diagonal of a square s where s is an side of the cubealso the face diagonal of a cube diagonal of a cube s where s is an edge of the cube diagonal of a rectangle  find either the length and the width or one dimension and the proportion of one to the other deluxe pythagorean theorem used to find length of the main diagonal of a rectangular solidd2 = x2 + y2 + z2, where x, y, and z are the sides of the rectangular solid and d is the main diagonalno formula approach: find diagonal of the bottom face and use this as the base leg of another right triangle (see page 4.35) similar triangles all corresponding angles are equal and their corresponding sides are proportionif two similar triangles have corresponding side lengths in ratio a:b, then their ares will be in ratio a2:b2 area of an equilateral triangle with a side length of S area of a right triangle (1/2) Hypotenuse x height from hypotenuse radius any line segment that connects the center point to a point on the circlehalf the distance across a circle chord any line segment that connects two points on a circle diameter any chord that passes through the center of the circle2 x r circumference d2 rrevolution (i.e. a full turn of a spinning wheel) area of a circle r2  arc length corresponds to the crustfirst find the circumference of the circlethen, use the central angle to determine what fraction the arc is of the entire circlemultiple fraction and C perimeter of a sector corresponds to a slice of pizzaformed by the arc and two radii (sum arc length and two radii) area of a sector of a circle first, find the area of the entire circlethen, use the central angle to determine what fraction of the entire circle is represented by the sectormultiply fraction and area central angle an angle whose vertex lies at the center point of a circle inscribed angle has its vertex on the circle itselfequal to half of the arc it intercepts inscribed triangles a triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circleif one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle surface area of a cylinder 2 r2 + 2 rh volume of a cylinder r2h three properties of intersecting lines interior angles form a circle; sum of these angles is 360 degreesinterior angles that combine to form a line sum to 180 degrees (supplementary angles)angles found opposite each other where these two lines intersect are equal (vertical angles) exterior angles of a triangle equal to the sum of the two non-adjacent (opposite) interior angles of the triangle parallel lines cut by a transversal all acute angles are equalall obtuse angles are equalany acute angle is supplementary to any obtuse angle slope of a line rise over run 4 types of slopes positivenegativezero (----)undefined slop (l) intercept (x- and y-intercept) a point where a line intersects a coordinate axisx-intercept is the point on the line at which y=0y-intercept is the point on the line at which x=0to find x-intercept, plug in 0 for yto find y-intercept, plug in 0 for x slope-intercept equation y = mx + b horizontal lines y = some number vertical lines x = some number finding distance between 2 points use pythagorean theorem Data Sufficiency Answer Choices (A) Statement 1 ALONE is sufficient, but statement 2 is NOT sufficient(B) State 2 ALONE is sufficient, but statement 1 is NOT sufficient(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient(D) EACH statement ALONE is sufficient(E) Statement 1 and 2 TOGETHER are NOT sufficient The Data Sufficiency Process Separate additional info from the actual questiondetermine whether the question is Value or Yes/NoDecide exactly what the question is askingUse the Grid to Evaluate the statementsADBCE General Geometry Approach Draw or redraw the figurefill in the given informationIdentify the wanted elementInfer from the givensFind the wanted element Unspecified number - picking numbers (e.g. If the length of the side of a cube decreases by one-half, by what percentage will the volume of the cube decreases?) Infer cube has a side of 2 unites ("smart" number bc it is divisible by 2 - the denominator of one-half)volume = 2 x 2 x 2 = 8if cube decreases by one-half, its new length is 2 - .5 (2) = 1 unitits new volume = 1 x 1 x 1 = 1Percent decreases = change/original(8-1)/8 = 7/8 = 0.875 or 87.5% similar triangles (2) any time two triangles each have a right angle and also share an additional right angle, they will be similar maximum area of a quadrilateral of all quadrilaterals with a given perimeter, the square has the largest areaof all quadrilaterals with a given area, the square has the minimum perimeter maximum area of a parallelogram or triangle if you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each otherrule holds for rhombuses as well key questions about parabola: (1) how many times does the parabola touch the x-axis? (2) If the parabola does touch the x-axis, where does it touch? In other words, the parabola touches the x-axis at those values of x that makes f(x)=0You can solve for zero by factoring and solving the equation directly. You might plug in points and draw the parabolaUse quadratic formula to quickly tell how many solutions the equation has by looking at the discriminate (b^2 = 4ac) (1) If b^2 - 4ac > 0, two roots --> parabola crosses the x-axis twice and has two x-intercepts(2) If b^2 - 4ac = 0, produce one root, parabola touches the x-axis once and has just one x-intercept(3) If b^2 - 4ac < 0, square root operation cannot be performed, produces no roots, parabola never touches the x-axis (it has no x-intercept) Perpendicular Bisectors (rare on GMAT) forms a 90 degrees angle with the segment and divides the segment exactly in half the perpendicular bisector has the negative reciprocal slope of the line segment it bisects(1) Find the slope of segment AB(2) Find the slope of the perpendicular bisector of AB(3) Find the midpoint of ABmidpoint between point A(x1, y1) and point B(x2, y2) is ( , )(4) Put the information together. The Intersection of Two Lines  ex. at what point does the line represented by y-4x-10 intersect the line represented by 2x+3y=26 replace y in the second equation with 4x-10 and solve for xsolve for y Authorjeffhn90 ID193654 Card Setgmat geometry Descriptionbasic geometry flashcards Updated2013-03-14T02:57:58Z Show Answers