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Gravity
Terms in this set (58)
what is the formula for the AREA of any triangle?
1/2 base * height (perpendicular)
what is the side length formula for a RIGHT triangle?
what are the proportions of a 30-60-90 RIGHT triangle?
30: x
60: x√3
90: 2x
what are the proportions of a 45-45-90 RIGHT ISOSCELES triangle?
45: x
45: x
90: x√2
what is the length of the longest side of a triangle?
the hypotenuse
what is the definition of an ISOSCELES triangle?
a triangle that has AT LEAST two sides and two angles that are equal
what shapes form from the split of an ISOSCELES triangle?
two separate 30-60-90
RIGHT triangles that are symmetrical
what shapes form from a square that is split?
two 45-45-90 triangles; the 45-45-90 formula is used to determine the DIAGONAL of a square (its longest length)
what is the definition of an EQUILATERAL triangle?
a triangle with sides of equal length and all equal angles that measure 60 degrees
what is the AREA of an EQUILATERAL triangle?
what do all GMAT pimps know about EQUILATERAL triangles?
any piece of information is enough to determine everything about that triangle. I.e. if you know the perimeter, you can determine the area. if you know the area, you can find the side lengths.
what is the 'Third Side Rule' of a triangle?
The 3rd side of a triangle is always > than the difference of the other two sides and < the sum of the those other two sides.
If 7 and 10 are the lengths of two sides of a triangular region, which of the following can be the length of the third side?
a) 2 b) 8 c) 17
b) 8 (greater than 10-7 = 3 and less than 7+10 = 17)
GMAT pimps know what about EXTERIOR ANGLES?
Supplementary angles adjacent to interior angles will always equal the sum of the other two interior angles.
What are the common side ratios (Pythagorean triples) of RIGHT triangles?
3:4:5
5:12:13
7:24:25
8:15:17
9:40:41
(or any proportion/multiple of the above)
what do GMAT pimps know when they see the term 'not drawn to scale' next to figures?
these figures are intentionally manipulated to sucker you into making false assumptions. "trust the information, not the representation."
what do SIMILAR triangles have in common?
all the angles of the two triangles are the same. thus, the PROPORTION of corresponding sides and height is the same too.
what are the 3 standard ways to identify SIMILAR triangles?
1) match at least two sets of angles
2) constant ratio of corresponding sides (all 3 sides)
3) constant ratio for 2 sets of SIDES -and- if the single ANGLE measurement between them is the same
what is the AREA of a PARALLELOGRAM?
base x height
what is the AREA of a TRAPEZOID?
1/2 (base 1 + base 2) x height
what are the properties of the DIAGONALS of a SQUARE?
1. equal in length
2. intersect at 90 degrees
3. bisect each other
what are the properties of the DIAGONALS of a RECTANGLE?
1. equal in length
2. bisect each other
what are the properties of the DIAGONALS of a RHOMBUS?
1. intersect at 90 degrees
2. bisect each other
what are the properties of the DIAGONALS of a PARALLELOGRAM?
1. bisect each other
what is the formula for the sum of the INTERIOR angles of any POLYGON?
(n-2) x 180
n= number of sides of the polygon
What is the measure of each angle of a regular PENTAGON?
5 sides - 2 = 3
3 x 180 = 540
540/5 = 105 degrees for each angle
what is the formula for the VOLUME of a rectangular solid?
Length x Width x Height
what is the formula for the SURFACE area of a rectangular solid?
2 (LW + LH + WH)
in roman numeral problems, what is a good strategy to eliminate answers?
find the numeral that appears the most in the answer choices and test to see if it's true or not; from there eliminate answers
what is the COMBINED WORK formula?
1/A + 1/B = 1/T
A = time for unit 1
B = time for unit 2
T = time together
what is the SHORTCUT combined work formula?
AB / (A+B) = T
A = time for unit 1
B = time for unit 2
T = time together
what is the MIDPOINT formula for two coordinates?
(x + x/2), (y + y/2)
what's the relationship between QUADRILATERALS and TRIANGLES on the GMAT?
A square can be split into 2 isosceles right triangles. The diagonal of a rectangle likewise form right triangles. A right triangle is often needed to find the height of a trapezoid or parallelogram.
what is a CHORD?
line that connects any 2 points on a circle. the diameter is an example of a chord.
what is an ARC?
any portion of the circumference on a circle
what is a CENTRAL ANGLE?
any angle whose vertex (point of origin) is at the center of the circle
what is an INSCRIBED ANGLE?
any angle whose vertex (point of origin) is on the circumference of the circle
what is a SECTOR?
a portion of the circle defined by two radii and an arc carved by a central angle (like a big piece of pizza)
what is a TANGENT?
a line that touches a circle at only one point on a circle. the tangent is PERPENDICULAR to the RADIUS at the point of tangency
what is the formula relating ARCS and CENTRAL ANGLES and CIRCUMFERENCE of a circle?
(central angle measurement/360) = (minor arc length/total circumference)
what is the relationship between INSCRIBED ANGLES and CENTRAL ANGLES?
INSCRIBED ANGLES that cut out the same arc will always be equal in measurement. INSCRIBED ANGLES that cut out the same arc as a CENTRAL ANGLE will always be 1/2 the measurement of the CENTRAL ANGLE.
what is the greatest possible distance within a rectangular box?
In a rectangular box (between 2 corners)"
sqrt( L^2 + W^2 + H^2)
if the SLOPE has a POSITIVE value, which direction will the line be pointing?
from the bottom left to the top right quadrants, pointing up (like when you tagged Melissa's FOB Korean friend after 'tea')
if the SLOPE has a NEGATIVE value, which direction will the line be pointing?
from the top left to the bottom right quadrants, pointing down (like when you drink too much)
in the equation y = mx + b, b represents what?
y-intercept (intersection with the y-axis)
in the equation y = mx + b, m represents what?
slope
slope represents what in coordinate geometry?
change in y-coordinate/change in x-cordinate
what is the formula for the x-intercept?
-(b/m)
what is the DISTANCE formula between two points in coordinate geometry?
find the x distance and y distance and square them to find the distance in squared form (don't forget to add a root over the squared value)
parallel lines in coordinate geometry always have _______ slope
equal
-all the lines below have a slope of 1
y = x + 2
y = x - 2
y = x
slopes of perpendicular lines always have a product = to what?
1
y = -(1/2)x + 2 -and- y = 2x + 4 are perpendicular (-1/2) and (2) are negative reciprocals of each other and have a product = -1
what is the minimal information you need to determine the equation of a line?
a) any 2 points on a line
b) 1 point on a line and the slope
c) 1 point on the line and the slope or equation of a line perpendicular to that line
d) 1 point on the line and the slope or equation of a line parallel to that line
We are given the following information about the figure on the back of this flashcard:
1) The lengths of AC and AB.
2) The fact that DEF has the same area as the shaded region.
3) The ratio DF:EF is equal to the ratio AC:BC.
In the figure above, the length of AC is 12 and the length of AB is 15. The area of right triangle DEF is equal to the area of the shaded region. If the ratio of the length of DF to the length of EF is equal to the ratio of the length of AC to the length of BC, what is the length of EF?
(ctnd from previous flashcard):
1) In right triangle ABC, AB = 15 and AC = 12. So right triangle ABC is a multiple of the 3:4:5 right triangle with each side of right triangle ABC being 3 times the corresponding member of the 3:4:5 ratio. We can see that AB = 15 = 3 × 5 and AC = 12 = 4 × 3, so BC = 3 × 3 = 9.
2) Since the area of triangle DEF is equal to the area of the shaded region, and triangle DEF and the shaded region together make up triangle ABC, the area of triangle DEF is half the area of triangle ABC. Thus, the area of right triangle ABC is (1/2)(9)(12) = 54. Therefore, the area of triangle DEF is (1/2)(54), or 27.
3) Now to incorporate the final piece of information: the ratios DF:EF and AC:BC are equal. Now that we have calculated the actual values of the sides of ABC, we know the ratio of the length of AC to the length of BC is 12:9, which reduces to 4:3. So the ratio of the length of DF to the length of EF is also 4:3. We can let the length of DF be 4x and the length of EF be 3x. (Notice that DEF is also a 3:4:5 triangle.)
The area of triangle DEF is 27. Since DEF is a right triangle just like ABC, the area of triangle DEF is also (1/2)(EF)(DF). So the area of triangle DEF is (1/2)(4x)(3x) = 6x^2. Simplifying, 6x^2 = 27, and x^2 = 27/6, x^2 = 9/2, so x= (radical 9/2) or x = 3 / (radical 2).
Remember that the length of EF (the side length we are asked to solve for) is 3x, and x = 3 / (radical 2), so multiply 3 (3 / radical 2) = 9/ (radical 2)
None of the answer choices is 9 / (radical 2) , so as a final step, let's try to rewrite so the denominator will not contain a radical. We will do this by multiplying the numerator and denominator of by radical 2. Doing so, we end up with 9 (radical 2) / (radical 2) x (radical 2) = 9 (radical 2) / radical 4, which equals 9 (radical 2) / 2 .The length of EF is 9 (radical 2) / 2, so Answer Choice (C) is correct.
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With Isosceles triangles, angles that are opposite sides of equal length...
...have angle measurements that are also equal. (CAT #3, Q18)
With Isosceles triangles, angles that are opposite sides of equal length...
...have angle measurements that are also equal. (CAT #3, Q18, and CAT #5, Q14)
What is important to keep in mind as it relates to a triangle's angle measurements and and supplementary angle measurements?
the sum of both the triangle's angle measurements and any two supplementary angle measurements will both equal 180.
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