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GMAT: Geometry
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Gravity
From book 5 of the Manhattan GMAT series.
Terms in this set (33)
Polygons
A closed shape formed by line segments.
2-D shapes: interior angles, perimeter, area
3-D shapes: surface area, volume
Quadrilaterals
Parallelogram: opposite sides and angles are equal.
Trapezoid: one pair of opposite sides is parallel.
Rectangle: all angles are 90 degrees and opposite sides are equal.
Rhombus: all sides are equal; opposite angles are equal.
Square: all angles are 90 degrees; all sides are equal.
*squares are both a rectangle and a rhombus.
Polygons & Interior Angles
Sum of interior angles of a polygon = (n-2) x 180, where n is the number of sides.
Corners of a polygon known as vertices (vertex singular).
Can cut a polygon into triangles, count the number of triangles, and multiply by 180.
Triangle = 3 sides = 180 degrees
Quadrilateral = 4 sides = 360 degrees
Pentagon = 5 sides = 540 degrees
Hexagon = 6 sides = 720 degrees
Polygons: Perimeter & Area
Perimeter: sum of lengths of all sides.
Area: space inside a polygon
Triangle: (b*h)/2, where h is a line perpendicular to the base
Rectangle: L x W
Trapezoid: ((base1 + base2) x height)/2
Parallelogram: base x height
Rhombus: (diagonal1 x diagonal2)/2; diagonals of a rhombus always perpendicular.
If you forget an area formula, cut a shape into rectangles and right triangles and sum those areas.
3-D: Surface Area & Volume
Surface Area: the sum of all areas of all faces.
Volume: L x W x H
Remember when fitting 3-D objects into other 3-D objects to take into account the specific dimensions of each object. Dimensions matter when seeing if objects fit into each other without gaps.
Triangles
Most commonly tested polygon.
Right triangles: you can find the length of a third side given two sides by using the Pythagorean theorem. Essential for solving problems with other polygons.
30-60-90 & 45-45-90 right triangles: only need the length of one side to find the other two sides.
Triangles: Angles
1. The sum of all angles is 180 degrees.
2. Angles correspond to their opposite sides (largest angle opposite largest side).
- if two sides are equal, opposite angles are also equal (isosceles triangle).
Be ready to redraw to easily see the triangle. Use hash marks to show equal lengths/angles.
Sides of a Triangle
Triangle Inequality Law: the sum of any two sides of a triangle must be greater than the third side.
The sum of two sides cannot equal the third side.
If given two sides of a triangle, the length of the third side must lie between the difference and the sum of the other given sides.
Pythagorean Theorem & Common Right Triangles
To find the legs of a right triangle:
A^2 + B^2 = C^2
Common right triangles - memorize!
3-4-5 leads to 6-8-10, 9-12-15, 12-16-20
5-12-13 leads to 10-24-26
8-15-17
Isosceles Triangle and the 45-45-90 Triangle
Isosceles - 2 sides & 2 opposite angles are equal.
45-45-90 = isosceles right triangle
*lengths have a specific ratio (memorize)
*leg leg hypotenuse
*1 : 1 : sq root of 2
x : x : x
sq root of 2
This triangle is exactly half of a square. If you are given the diagonal of a square, you can use the above ratio to find the length of a side.
Equilateral & 30-60-90 Triangles
Equilateral = all 3 sides and angles (60 degrees) are equal.
- they are two 30-60-90 triangles put together
Lengths of a 30-60-90 triangle follow a specific ratio (memorize):
*short leg : long leg : hypotenuse
*1 : sq root of 3 : 2
x : x
sq root of 3 : 2x
The side of an equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle. The height of an equilateral triangle is the same as the long leg of a 30-60-90 triangle.
Diagonals of other Polygons
Right triangles can find the diagonals of squares, cubes, rectangles and rectangular solids.
Diagonal of a square = s*sq root of 2; s is a side
Diagonal of a cube = S*sq root of 3; S is a side
To find the diagonal of a rectangle, you must know either the length and the width OR one of those and the proportion. To find the diagonal of a rectangular solid, use the pythagorean theorem twice - once for the bottom leg of the diagonal and the second time to find the diagonal.
D^2 = X^2 + Y^2 + Z^2
Similar Triangles
Triangles are similar if all their corresponding angles are equal and their corresponding sides are in proportion. Only need to know if 2 pairs of angles are congruent.
If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a^2:b^2.
- for similar solids, their volumes will be on ratio a^3:b^3.
The principle holds for any similar figures (other polygons).
Triangles & Area II
Since a triangle has one area, the area must be the same regardless of the side chosen as the base.
The area of an equilateral triangle with a side of length S equals
(S^2 x sq root of 3) / 4
Circles & Cylinders
Circle - set of points in a plane that are equidistant from a fixed center point. They have 360 degrees.
Radius - any line segment that connects two points on a circle.
Chord - any line segment that connects two points on a circle.
Diameter - any chord that passes through the center of the circle.
GMAT tests:
1. Circumference
2. Area of whole and partial circles
3. Surface area
4. Volume of cylinders
Circumference of a Circle
Circumference = distance around a circle
C = 2(pi)r
Pi can usually be approximated as 3 or 22/7; knowing pi is approximately 3 can help rule out small or large answers.
A full revolution of a spinning wheel equals the wheel going around once. A point on the edge of a wheel travels one circumference in one revolution.
Circles: Arc Length & Perimeter of a Sector
Portion of the distance on a circle is an arc. Find the arc length by determining what fraction the arc is of the entire circumference.
The boundaries of a sector of a circle are formed by the arc and two radii. If you know the radius and central (or inscribed) angle, you can find the perimeter of the circle.
Area: Circles & Sectors
Area = (pi)r^2
Area of a sector can be found by determining the fraction of the entire area that the sector occupies. To determine this fraction, look at the central angle of the sector.
Inscribed vs. Central Angles & Inscribed Triangles
Central angle: an angle whose vertex lies at the center point of a circle.
Inscribed Angle: an angle with a vertex on the circle.
Inscribed Triangle: all vertices of the triangle are points on a circle. If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle MUST be a right triangle.
Cylinders: Surface Area and Volume
SA = 2 circles + rectangle = 2(pi
r^2) + 2pi
r*h
V = pi
r^2
h
To find SA and V, you only need the radius and height of the cylinder.
Lines & Angles
A straight line is the shortest distance between two points. A line measures 180 degrees.
Parallel lines never intersect. Perpendicular lines intersect at 90 degree angles.
2 major relationships to know:
1. The angles formed by any intersecting lines.
2. The angles formed by parallel lines cut by a transversal.
Intersecting Lines
1. The interior angles formed by intersecting lines form a circle, so the sum of these angles is 360 degrees.
2. The interior angles that combine to form a line sum to 180 degrees. These are termed supplementary angles.
3. Angles found opposite each other where 2 lines intersect are equal. These are vertical angles.
Note that these rules apply to more than two lines that intersect at a point.
Exterior Angles & Parallel Lines
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles of the triangle. This is frequently tested on the GMAT!
When parallel lines are cut by a transversal, all acute angles are equal and all obtuse angles are equal. Any acute angle is supplementary to any obtuse angle (adds up to 180 degrees).
Coordinate Plane & Slope of a Line
Points in a plane are identified using an ordered pair of numbers. The point (0,0) where the axes cross is called the origin.
Slope of a line = rise/run = (y1 - y2)/(x1 - x2)
A line has a constant slope.
Four types of slope:
Positive - upward from left to right
Negative - downward from left to right
Zero - horizontal line
Undefined - vertical line
Intercepts of a Line
A point where a line intersects a coordinate axis is called an intercept.
The x-intercept is the point on the line at which y=0. The y-intercept is the point on the line ate which x=0.
The find x-intercepts, plug 0 in for y. To find y-intercepts, plug 0 in for x.
Slope-Intercept Equation
All lines can be written as equations: y = mx + b
m = slope of the line
b = y-intercept of the line
Linear equations represent lines on the coordinate plane. They often look like Ax + By = C where A, B, and C represent numbers. Linear equations never involve exponentials, roots or xy.
Horizontal lines are y = some number
Vertical lines are x = some number
How to Find the Equation of a Line
If given two coordinates or one coordinate and the y-intercept:
1. Find the slope by calculating rise over run. Subtract the coordinates in the same order.
2. Plug the slope in for m.
3. Solve for b by plugging in one of the coordinates into the equation.
4. Write the equation in the form of y = mx + b.
The Distance between 2 Points on a Coordinate Plane
The distance between any two points in a plane can be calculated by using the Pythagorean Theorem.
1. Draw a right triangle connecting the points.
2. Find the lengths of the two legs by calculating the rise and the run.
3. Use the Pythagorean Theorem to calculate the length of the diagonal (distance between two points).
Positive & Negative Quadrants
Quadrant I: x & y positive
Quadrant II: x negative, y positive
Quadrant III: x & y negative
Quadrant IV: x positive, y negative
To determine which quadrants a line passes through:
1. Rewrite the line in form y = mx + b.
2. Sketch the line.
OR
1. Find two points on a line by setting x&y equal to zero.
2. Sketch the line using the x & y intercepts.
Perpendicular Bisectors
The perpendicular bisector of a line segment forms a 90 degree angle with the segment and divides the segment exactly in half.
1. Find the slope of the segment.
2. Find the slope of the perpendicular bisector of the segment. It is the negative, reciprocal slope of the segment.
3. Find the midpoint of the segment, which is the average distance between the x and y of the endpoints.
4. Find the b-intercept of the line using the midpoint of the segment.
5. Fill in the equation y = mx + b.
Intersection of Two Lines
When two lines intersect on a coordinate plane, at the point of intersection, BOTH equations representing the lines are true for that intersecting point.
Find the point of intersection using a system of equations:
1. Replace the y of one equation with the y equivalent from the other equation and solve for x.
2. Use that x to solve for y in one of the original equations.
If two lines are parallel, no x and y satisfy both equations at the same time.
If two equations represent the same line, then they intersect at an infinite number of points.
Maximum Area of Polygons
Quadrilateral:
- with a fixed perimeter, the SQUARE has the largest area.
- of all quadrilaterals with a given area, a square has the minimum perimeter.
- a regular polygon with all sides equal will maximize area for a given perimeter and minimize perimeter for a given area.
Triangle/Parallelogram:
- if given 2 sides, to maximize the area, make those sides the base and height and make the angle between 90 degrees.
- you can maximize the area of a rhombus with a given side length by making it a square.
Function Graphs & Quadrilaterals
Parabola: curved graph, quadratic function
F(x) = ax^2 + bx + c
- positive a graph curves up
- negative a graph curves down
- large |a| narrow curve
- small |a| wider curve
1. How many times does the parabola touch the x-axis?
2. If it does touch, where does it intercept?
You can find the x-intercept by using the quadratic formula:
x = -b +\- sqroot(b^2 - 4ac) / 2a
1. If the discriminant (b^2 - 4ac) > 0, there are two intercepts.
2. If the discriminant = 0, there is one intercept.
3. If the discriminant < 0, there are no intercepts.
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