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Terms in this set (63)
A ______ is a special type of parallelogram that is both a rectangle and a rhombus
Square
Sum of the interior angles of a polygon =
This sum is always 180 degrees times 2 less than N (the number of sides)
Triangle , 3, 180
Quadrilateral, 4, 360
Pentagon, 5, 540
Hexagon, 6 , 720
Area of a Trapezoid =
{(Base 1 + Base 2) x height } / 2
Area of any parallelogram =
Base x Height
Area of a Rhombus =
( Diagonal 1 X Diagonal 2 ) / 2
Volume =
Length x width x height
How many books, each with a volume of 100 inches cubed, can be packed in to a crate with a volume of 5,000 in3?
The answer is not 50 books... Without knowing the exact shapes of all the books, you cannot tell whether they would all fit into the crate. Remember, when you a fitting 3-dimensional objects into other 3-dimenstional objects, knowing the respective volumes is not enough. You must know the specific dimensions ( L, W, H) of each object to determine whether the objects can fit without leaving gaps.
Triangles: Angles correspond to their opposite sides. This means that the largest angle is opposite to the longest side, while the smallest angle is opposite to the shortest sides
if two sides are equal, their opposite angles are also equal. these are called Isosceles triangles
The sum of any two sides of a triangle must be...
Greater than the third side
Note that the length cannot be as small as we wish either
it must be greater than the difference between the lengths of the other two sides.
If you are given two sides of triangle, the length of the third side must lie between the difference and the sum of the two given sides. Triangle has sides length 3 and 4
Third side has to be between 1 and 7
Common right triangles
3, 4, 5 = 6, 8, 10, 9, 12, 15, 12, 16, 20
5, 12 , 13 = 10, 24, 26
5, 15, 17
The lengths of the legs of every 45 - 45 - 90 triangle have a specific ratio, which you must memorize:
45 - 45 - 90
leg - leg - hypotenuse
1 : 1 : Square root of 2
x : x : X Square root of 2
Reason 45 - 45 - 90 is important?
this triangle is exactly half of a square! If given the diagonal of a square, use the 45-45-90 ratio to find out the length of a side of the square.
The lengths of the legs of every 30-60-90 triangle have the following ratio:
30 - 60 - 90
Short leg - Long leg - Hypotenuse
1 : SQRT 3 : 2
x : X SQRT 3 : 2x
The diagonal of a square can be found using this formula
d = s SQRT 2
The main diagonal of a cube can be found using this formula
d = s SQRT 3
Deluxe Pythagorean theorem:
d^2 = x^2 + y^2 + z^2
Once you find that 2 triangles have 2 pairs of equal angles
you know that the triangles are similar. if 2 sets of angles are congruent, then the third angles must be congruent, since the sum of the angles in any triangle is 180 degrees
if two similar triangles have corresponding side lengths in ratio a:b, then
their areas will be in ratio a^2:B^2
The principle holds true for any similar figures: quadrilaterals, pentagons, etc. For similar solids with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3
page 37 Geometry book
area of an equilateral triangle with a side of length S is equal to
1/2 (S) (S Sqrt3 / 2) = S^2 SQRT 3 / 4
how to find circumference and arc length
find the circumference, then use the angle for example 60 degrees and find which percent of the circle is the arc. 60 / 360 = 1/6. if R = 12, C = 24pi, arc length = 4pi
inscribed angle is _____ of central angle
1/2 p 51 geometry book An inscribed angle is equal to half of the arc it intercepts
inscribed triangles
if one of the sides of an inscribed triangle is the diameter of the circle, then the triangle must be a right triangle
Conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides (thereby splitting the circle in half.
...
surface area of a cylinder =
2 circles + rectangle = 2(piR^2) + 2piRH
volume of a cylinder =
piR^2*H
The interior angles formed by intersecting lines for a _____
Circle, so the sum of these angles = 180 degrees.
supplementary angles =
interior angles that combine to form a line sum to 180 degrees
Vertical angles
angles found opposite each other where these two lines intersect are equal
An exterior angles of a triangle is =
equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle.. page 60 geometry book
The sum of the interior angles depends on the number of sides in the polygon. what is the formula?
(n-2) * 180
64 - 225 - 289
8 - 15 - 17 Triangle
Isosceles Triangles and the 45 - 45 - 90
1 - 1 -SQRT2
if given the diagonal of a square, you can use the 45 - 45 - 90 ratio to find the length of a side of the square
Diagonal is X SQRT 2
30 - 60 - 90
1x : SQRT 3x : 2x
Equilateral triangles and the 30 - 60 - 90. (Equilateral is made up of 2 30, 60 , 90s
main diagonal of a cube =
d = s SQRT 3 where s is an edge of the cube
What is the measure of an edge of a cube with a main diagonal of length SQRT 60
SQRT 20
Similar triangles
triangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion.
Once you find that 2 triangles have 2 pairs of equal angles, you know that the triangles are similar. If 2 sets of angles are congruent, then the third set of angles must be congruent, since the sum of the angles in any triangle is 180
If two similar triangles have corresponding side lengths in ratio A:B then their areas will be in ratio___
A^2: B^2
area of an equilateral triangle
S^2 ( SQRT 3) / 4
Diagonal of a rectangular box
X^2 + Y^2 + Z ^2 = D^2
x , y , z = sides and height D = diagonal
The interior angles formed by intersecting lines form a circle, so the sum of these angles is
360 degrees
interior angles that combine to form a line sum to
180 degrees - these are termed supplementary angles.
angles found opposite each other where these two lines intersect are
equal. These are called vertical angles.
An exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle
The is FREQUENTLY tested
parallel lines cut by a transversal
all the acute angles (less than 90 degrees) in the diagram are equal. likewise all obtuse (more than 90 but less than 180 degrees) are equal.
when you see a problem with two parallel lines
extend lines and label the acute and obtuse angles.
What is the slope - intercept form for a line with the equation 6x + 3y = 18
y = -2x + 6
y-intercept is (0,6) slope is -2
Horizontal lines are expressed in the form: y = some number
...
Vertical lines are express in the form x = some number
...
The distance between any two points in the coordinate plane can be calculated by using the Pythagorean Theorem
...
the y intercept is the point on the line at which
x = 0
Of all quadrilaterals with a given perimeter what has the largest area
square
Of all quadrilaterals with a given area what has the minimum perimeter
square
if you are given two sides of a triangle or parallelogram you can maximize the area by placing those two sides perpendicular to each other
...
Because angles BAD and ACD are right angles, the figure above is composed of three similar right triangles: BAD, ACD and BCA. Any time a height is dropped from the right angle vertex of a right triangle to the opposite side of that right triangle, the three resulting triangles have the same 3 angle measures. This means that they are similar triangles. (See your Strategy Guide if you want to explore this rule further!)
Question 2 on geometry practice problem on mgmat site.
basically just tip over the left triangle so it matches the triangle on the right and set up a proportion.
To solve for the length of side CD, we can set up a proportion, based on the relationship between the similar triangles ACD and BCA:
Is quadrilateral ABCD a rectangle?
(1) Line segments AC and BD bisect one another.
(2) Angle ABC is a right angle.
(1) INSUFFICIENT: The diagonals of a parallelogram bisect one another. Knowing that the diagonals of quadrilateral ABCD (i.e. AC and BD) bisect one another establishes that ABCD is a parallelogram, but not necessarily a rectangle.
(2) INSUFFICIENT: Having one right right angle is not enough to establish a quadrilateral as a rectangle.
(1) AND (2) SUFFICIENT: According to statement (1), quadrilateral ABCD is a parallelogram. If a parallelogram has one right angle, all of its angles are right angles (in a parallelogram opposite angles are equal and adjacent angles add up to 180), therefore the parallelogram is a rectangle.
The correct answer is C.
Is quadrilateral ABCD a rhombus?
(1) Line segments AC and BD are perpendicular bisectors of each other.
(2) AB = BC = CD = AD
(1) SUFFICIENT: The diagonals of a rhombus are perpendicular bisectors of one another. This is in fact enough information to prove that a quadrilateral is a rhombus.
(2) SUFFICIENT: A quadrilateral with four equal sides is by definition a rhombus.
The correct answer is D.
Is quadrilateral ABCD a square?
(1) ABCD is a rectangle.
(2) AB = BC
(1) INSUFFICIENT: Not all rectangles are squares.
(2) INSUFFICIENT: Not every quadrilateral with two adjacent sides that are equal is a square. (For example, you can easily draw a quadrilateral with two adjacent sides of length 5, but with the third and fourth sides not being of length 5.)
(1) AND (2) SUFFICIENT: ABCD is a rectangle with two adjacent sides that are equal. This implies that all four sides of ABCD are equal, since opposite sides of a rectangle are always equal. Saying that ABCD is a rectangle with four equal sides is the same as saying that ABCD is a square.
The correct answer is C.
when two similar three-dimensional objects are compared, the volume ratio will be what?
the volume ratio will be the cube of the length ratio. For example if it is given that the length of an edge of cube X is twice the length of an edge of cube Y, the length ratio for cube Y to cube X is 1/2. This therefore makes the volume ratio 1/2^3 = 1/8
How to find the radius of a circle inscribed inside a right triangle A, B, C
A+B - C / 2 if a circle x is inscribed in triangle, 6, 8, 10. R =
6+8 - 10 / 2 = 4 / 2 = 2 = R
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