Papgust's GMAT MATH FLASHCARDS directory
- papgust
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Really sorry for a long break. I will be posting flashcards on "Geometry", the last topic of these flashcards.
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- papgust
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Types of angles:
Figure 1:
1. Adjacent angles: Any 2 angles that share a common side separating the 2 angles and that share a common vertex.
E.g.: | 1 and | 2 are adjacent angles. [See Figure 1]
2. Vertical angles: Any 2 angles that are not adjacent angles. Vertical angles are EQUAL in measure.
E.g.: | 1 and | 3 are vertical angles. [See Figure 1]
3. Complementary angles: Any 2 angles whose sum is 90 degrees. Complementary angles NEED NOT be adjacent to each other.
4. Supplementary angles: Any 2 angles whose sum is 180 degrees.
Figure 2:
L and M are parallel lines. T is a traversal.
5. Corresponding Angles: Angles that appear to be in the same relative position in each group of four angles. Corresponding angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 1 and | 5 are corresponding angles. [See Figure 2]
6. Alternate Interior Angles: Angles within the lines being intersected, on opposite sides of the traversal, and are not adjacent. Alternate interior angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 4 and | 6 are alternate interior angles. [See Figure 2]
7. Alternate Exterior Angles: Angles outside the lines being intersected, on opposite sides of the traversal, and are not adjacent. Alternate exterior angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 1 and | 7 are alternate interior angles. [See Figure 2]
8. Consecutive Interior Angles: Angles are same-side interior angles. Consecutive interior angles are SUPPLEMENTARY when two parallel lines are cut by a traversal.
E.g.: | 4 and | 5 are consecutive interior angles. [See Figure 2]
9. Consecutive Exterior Angles: Angles are same-side exterior angles. Consecutive exterior angles are SUPPLEMENTARY when two parallel lines are cut by a traversal.
E.g.: | 1 and | 8 are alternate interior angles. [See Figure 2]
Figure 1:
1. Adjacent angles: Any 2 angles that share a common side separating the 2 angles and that share a common vertex.
E.g.: | 1 and | 2 are adjacent angles. [See Figure 1]
2. Vertical angles: Any 2 angles that are not adjacent angles. Vertical angles are EQUAL in measure.
E.g.: | 1 and | 3 are vertical angles. [See Figure 1]
3. Complementary angles: Any 2 angles whose sum is 90 degrees. Complementary angles NEED NOT be adjacent to each other.
4. Supplementary angles: Any 2 angles whose sum is 180 degrees.
Figure 2:
L and M are parallel lines. T is a traversal.
5. Corresponding Angles: Angles that appear to be in the same relative position in each group of four angles. Corresponding angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 1 and | 5 are corresponding angles. [See Figure 2]
6. Alternate Interior Angles: Angles within the lines being intersected, on opposite sides of the traversal, and are not adjacent. Alternate interior angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 4 and | 6 are alternate interior angles. [See Figure 2]
7. Alternate Exterior Angles: Angles outside the lines being intersected, on opposite sides of the traversal, and are not adjacent. Alternate exterior angles are EQUAL when two parallel lines are cut by a traversal.
E.g.: | 1 and | 7 are alternate interior angles. [See Figure 2]
8. Consecutive Interior Angles: Angles are same-side interior angles. Consecutive interior angles are SUPPLEMENTARY when two parallel lines are cut by a traversal.
E.g.: | 4 and | 5 are consecutive interior angles. [See Figure 2]
9. Consecutive Exterior Angles: Angles are same-side exterior angles. Consecutive exterior angles are SUPPLEMENTARY when two parallel lines are cut by a traversal.
E.g.: | 1 and | 8 are alternate interior angles. [See Figure 2]
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- papgust
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Lines segments in Triangles:
* ALTITUDES:
Altitudes are the perpendicular segments from a vertex to the opposite sides. The three lines containing the altitudes intersect in a single point, which may or may not be inside the triangle.
* MEDIANS:
A Median is the line segment drawn from a vertex to the mid-point of its opposite side. The three medians meet in one point inside the triangle.
* ANGLE BISECTOR:
Angle bisector is a segment drawn from a vertex that bisects the vertex angle. The three angle bisectors meet in one point inside the triangle.
TAKEAWAY:
Altitude drawn from the vertex angle can be proven to be a median as well as an angle bisector in an ISOSCELES triangle.
* ALTITUDES:
Altitudes are the perpendicular segments from a vertex to the opposite sides. The three lines containing the altitudes intersect in a single point, which may or may not be inside the triangle.
* MEDIANS:
A Median is the line segment drawn from a vertex to the mid-point of its opposite side. The three medians meet in one point inside the triangle.
* ANGLE BISECTOR:
Angle bisector is a segment drawn from a vertex that bisects the vertex angle. The three angle bisectors meet in one point inside the triangle.
TAKEAWAY:
Altitude drawn from the vertex angle can be proven to be a median as well as an angle bisector in an ISOSCELES triangle.
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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An equiangular quadrilateral DOES NOT have to be equilateral.
An equilateral quadrilateral DOES NOT have to be equiangular. [--- Unlike Equilateral Triangle --]
An equilateral quadrilateral DOES NOT have to be equiangular. [--- Unlike Equilateral Triangle --]
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Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Regular Polygons:
* When a polygon is BOTH equilateral and equiangular.
* Sum of INTERIOR angles of a convex polygon with 'n' sides = (n-2) * 180
* Sum of EXTERIOR angles of a convex polygon = 360 degrees.
* When a polygon is BOTH equilateral and equiangular.
* Sum of INTERIOR angles of a convex polygon with 'n' sides = (n-2) * 180
* Sum of EXTERIOR angles of a convex polygon = 360 degrees.
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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How to prove a figure as PARALLELOGRAM?
5 ways:
* If both sides of opposite sides of a quadrilateral are EQUAL.
* If both pairs of opposite angles of a quadrilateral are EQUAL.
* If all pairs of consecutive angles of a quadrilateral are SUPPLEMENTARY.
* If one pair of opposite sides of a quadrilateral is both EQUAL and PARALLEL.
* If the diagonals of a quadrilateral BISECT each other.
TAKEAWAY:
A diagonal of a Parallelogram DIVIDES it into 2 congruent triangles.
5 ways:
* If both sides of opposite sides of a quadrilateral are EQUAL.
* If both pairs of opposite angles of a quadrilateral are EQUAL.
* If all pairs of consecutive angles of a quadrilateral are SUPPLEMENTARY.
* If one pair of opposite sides of a quadrilateral is both EQUAL and PARALLEL.
* If the diagonals of a quadrilateral BISECT each other.
TAKEAWAY:
A diagonal of a Parallelogram DIVIDES it into 2 congruent triangles.
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Isosceles Trapezoids:
* If the legs are EQUAL.
* Base Angles are EQUAL.
* Diagonals are EQUAL.
* Median of any trapezoid:
1. Is parallel to both bases.
2. Has length = 1/2 * (Sum of bases)
* If the legs are EQUAL.
* Base Angles are EQUAL.
* Diagonals are EQUAL.
* Median of any trapezoid:
1. Is parallel to both bases.
2. Has length = 1/2 * (Sum of bases)
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Regular Polygons:
* One point in its interior that is equidistant from its vertices is called the center of the regular polygon.
* An Apothegm is a line segment that goes from the center and is perpendicular to one of the polygon's sides.
Perimeter (regular n-gon) = n * s [n--> no of sides and s--> length of a side]
Area (regular n-gon) = 1/2 * a * p [a--> Apothegm length and p--> perimeter of regular n-gon]
* One point in its interior that is equidistant from its vertices is called the center of the regular polygon.
* An Apothegm is a line segment that goes from the center and is perpendicular to one of the polygon's sides.
Perimeter (regular n-gon) = n * s [n--> no of sides and s--> length of a side]
Area (regular n-gon) = 1/2 * a * p [a--> Apothegm length and p--> perimeter of regular n-gon]
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Similar Polygons:
* Two polygons with the same shape.
* When two polygons are similar, then the following MUST be true.
i. Corresponding angles are EQUAL.
ii. The ratios of pairs of corresponding sides must all be EQUAL.
* Two polygons with the same shape.
* When two polygons are similar, then the following MUST be true.
i. Corresponding angles are EQUAL.
ii. The ratios of pairs of corresponding sides must all be EQUAL.
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GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Similar Triangles:
If two triangles are similar, then the ratio of any two corresponding segments (such as altitudes, medians, angle bisectors) EQUALS the ratio of any two corresponding sides.
Example:
If ∆ QRS ~ ∆ TUV,
then QR/TU = RS/UV = QS/TV.
According to the theorem,
Length of altitude RA / Length of altitude UD = QR / TU
Length of median QB / Length of median TE = QR / TU
Length of bisector CS / Length of bisector FV = QR / TU
If two triangles are similar, then the ratio of any two corresponding segments (such as altitudes, medians, angle bisectors) EQUALS the ratio of any two corresponding sides.
Example:
If ∆ QRS ~ ∆ TUV,
then QR/TU = RS/UV = QS/TV.
According to the theorem,
Length of altitude RA / Length of altitude UD = QR / TU
Length of median QB / Length of median TE = QR / TU
Length of bisector CS / Length of bisector FV = QR / TU
Download GMAT Math and CR questions with Solutions from Instructors and High-scorers:
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- papgust
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Perimeters and Areas of Similar Triangles:
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangle.
i. If two similar triangles have a scale factor of a:b, the the ratio of their perimeters is a:b.
Example:
6/3 = 8/4 = 10/5 ==> 2/1 (or) 2:1 (Perimeter)
ii. If two similar triangles have a scale factor of a:b, then the ratio of their areas is a^2 : b^2.
Example:
Area of ∆ ABC / Area of ∆ DEF = 24 / 6 = 4 / 1 (or) 4:1
4:1 is nothing but a^2:b^2 (or) 2^2 : 1^2.
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangle.
i. If two similar triangles have a scale factor of a:b, the the ratio of their perimeters is a:b.
Example:
6/3 = 8/4 = 10/5 ==> 2/1 (or) 2:1 (Perimeter)
ii. If two similar triangles have a scale factor of a:b, then the ratio of their areas is a^2 : b^2.
Example:
Area of ∆ ABC / Area of ∆ DEF = 24 / 6 = 4 / 1 (or) 4:1
4:1 is nothing but a^2:b^2 (or) 2^2 : 1^2.
Download GMAT Math and CR questions with Solutions from Instructors and High-scorers:
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
https://www.beatthegmat.com/download-gma ... 59366.html
-----------
GO GREEN..! GO VEG..!
Daily Quote:
"Stop feeling sorry for the Butcher if you had to go Veg. The butcher can find another job but the poor animal cannot get back its life"
- surajgarg
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Pls don't end it, keep posting morepapgust wrote:Perimeters and Areas of Similar Triangles:
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangle.
i. If two similar triangles have a scale factor of a:b, the the ratio of their perimeters is a:b.
Example:
6/3 = 8/4 = 10/5 ==> 2/1 (or) 2:1 (Perimeter)
ii. If two similar triangles have a scale factor of a:b, then the ratio of their areas is a^2 : b^2.
Example:
Area of ∆ ABC / Area of ∆ DEF = 24 / 6 = 4 / 1 (or) 4:1
4:1 is nothing but a^2:b^2 (or) 2^2 : 1^2.
Is this rule only applicable for powers of 2? Thanks!papgust wrote:How to find REMAINDER for LARGE POWERS of numbers:
There are 2 ways to do so:
1. Pattern Method:
Example:
What is the remainder when 2^56 / 7 ?
Solution:
Remainder when 2^1 is divided by 7 is 2
Remainder when 2^2 is divided by 7 is 4
Remainder when 2^3 is divided by 7 is 1
Remainder when 2^4 is divided by 7 is 2 --> Repeats again.
The remainder repeats after 3 steps i.e. in the 4th step.
Now, Divide the power (or index) by 3 (no of steps after which remainder repeats) and compute a new remainder.
56 % 3 --> 2 (remainder)
Now, raise the base (2) to the power 2 (new remainder). 2^2 % 7 --> 4.
Thus, 4 is the remainder when 2^56 / 7.
2. Remainder Theorem Method: (NOT RECOMMENDED unless clear)
Example:
What is the remainder when 2^51 / 7 ?
Solution:
2^51 can be changed to (2^3)^17.
7 can be changed to (8-1) OR (2^3 - 1)
Substitute 'x' in place of 2^3,
x^17 / (x-1)
Remainder is f(1). Substitute 1 in 'x',
Remainder is 1.
Thus, 1 is the remainder when 2^51 / 7.
Can someone give an example? It's not completely clear to me..Thanks!papgust wrote:If ax^2+bx+c > 0, where a > 0,
Then
x DO NOT LIE between xL and xU
(xL and xU are lower and upper limits of x respectively)
If ax^2+bx+c < 0, where a > 0,
Then
x LIES between xL and xU
(xL and xU are lower and upper limits of x respectively)
- kvcpk
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Hi Samark,samark wrote:Can someone give an example? It's not completely clear to me..Thanks!papgust wrote:If ax^2+bx+c > 0, where a > 0,
Then
x DO NOT LIE between xL and xU
(xL and xU are lower and upper limits of x respectively)
If ax^2+bx+c < 0, where a > 0,
Then
x LIES between xL and xU
(xL and xU are lower and upper limits of x respectively)
Its simple. Let us take the equation as
x^2 -1 >0
Which values of x will satisfy this?
The lower limit of x is -1 and the upper limit of x is 1
Now the above rule says that x does not lie between -1 and 1.
Example take x=0 -> 0-1>0
-1 is not greater than 0.
if you take x=2, 4-1>0
3 is greater than 0.
Hence x should not lie between -1 and 1
Similarly, lets take x^2-1<0
Again upper and lower limits are -1 and 1
Now as per the rule above, x should lie between -1 and 1
Example take x=0 -> 0-1<0
-1 is less than 0.
if you take x=2, 4-1<0
3 is not less than 0.
Hope this helps!!