Geometry Flashcards
vertex (vertices)
a corner of a triangle
base
any side of a triangle
altitude
the line from the vertex and perpendicular to its opposite base
orthocenter
the joint point of three altitudes
median
the line from vertex to midpoint of its opposite base
centroid
joint point of three medians
when do we have 6 smaller triangles of equal area?
when we have the centroid
Ratio of vertex to centroid
2/3
Ratio of centroid to midpoint of triangle base
1/3
What is one of the formula to calculate area of triangle?
(P. r )/ 2
P = Perimeter of triangle
r = the radius of inscribed circle
What is one of the formula to calculate area of triangle?
(h.b) /2
h = altitude of triangle
b = base of triangle
What is one of the formula to calculate area of triangle?
(abc)/ 4R
a,b,c = bases of triangle
R = radius of circumscribed triangle
What is one of the formula to calculate area of triangle?
√s(s-a)(s-b)(s-c)
s = semi-perimeter of triangle
Two triangles are similar if
- their corresponding sides are in proportion
- square root of their area ratio equal to ratio of their bases
T or F. with corresponding sides in ratio of x/y, their volume will be in a ratio x^3 / y^3 for two similar solids
T
What does the two similar triangles theory extend?
to any 2-D figures
T or F. The shortest side is always opposite the largest interior triangle
False.
T or F. If two sides are equal, their opposite angles are equal
True.
T or F. The longest side is always opposite the largest interior triangle
True.
angle bisector
the line (from vertex) that divides a triangle angle into two equal angles.
incenter
the joint point of three angle bisectors
What is the incenter regarded as?
the center of inscribed circle in a triangle
a triangle is isosceles if:
- the triangle with two sides are equal in length
- the two angles are equal to each other
Formula to find the unknown base given the legs and altitude (in isosceles triangle).
How does this formula derive from?
B = 2. √(L^2 - A^2)
Pythagorean theorem
a triangle is equilateral if:
- all sides have same length, or
- all angles have 60 degrees
What is the formula to calculate area of equilateral triangle?
A = a^2 . (√3 / 4) a = length of the side
What is the formula to calculate the altitude of equilateral triangle?
h = a. (√3 / 2)
What is the formula to calculate the radius of circumscribed circle of equilateral triangle?
R = a. (√3 / 3)
What is the formula to calculate the radius of inscribed circle of equilateral triangle?
r = a. (√3 / 6)
For any point P within an equilateral triangle,
the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.
T or F. Isosceles right triangle has the largest area in any right triangle
True
What can we say about the distance from the midpoint of hypotenuse to each vertex in right triangle?
they are equal to each other
What are the ratio of sides in right triangle?
3:4:5, 5:12:13 or 8:15:17
What are the ratio of sides in isosceles right triangle?
1:1 : √2
What are the ratio of sides in 30-60-90 right triangle?
1: √3 : 2
What is one of the formula for the radius of the circle inscribed in a right triangle?
r = product of two sides (not the hypotenuse) / sum of three sides
What is one of the formula for the radius of the circle inscribed in a right triangle?
r = (sum of two sides - hypotenuse)/ 2
The sum of the interior angle of a polygon with n sides
(n-2) . 180
The perimeter of a polygon
the sum of the lengths of its side
T or F. If the diagonals of a rhombus are equal, then it must be a square
T
What is the ratio of diagonal of square to its side?
1.414 or √2
T or F. Of all quadrilaterals, with a given perimeter, the square has the smallest area
F.
T or F. Of all quadrilateral, with a given area, the square has the minimum perimeter
T
Calculation of an area of square from knowing the diagonal
d^2 / 2
Area of circle circumscribed square
(pi/ 2) . a^2
Area of circle inscribed the square
(pi/ 4) . a^2
what are the characteristics of sides and angles in rhombus?
all side are equal
opposite angles are equal
what is the characteristic of diagonals in rhombus?
their intersection forms a right angle
Area of rhombus
product of two diagonals divide by 2
Area of trapezoid
half of the product of height and sum of the base lengths
Isosceles trapezoid
trapezoid that has equal sides that are not parallel and both angles are same
What is the length of the median in trapezoid?
the average length of the bases
What is the length of minor arc in a circle, where x is the central angle of the arc in degree?
(2 π r x)/ 360
What is the area of the minor arc in a circle, where x is the central angle of the arc in degree?
x .π r^2/ 360
If the angle is not at the center, but at a different point on the circle then what is the angle of that minor arc?
angle = 90 L/ π r L = length of the minor arc
angle = 1/2 of angle of the minor arc at the center
T or F. It’s not an inscribed triangle if all of the vertices of the triangle are not points on the circle.
T.
T or F. If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be an equilateral triangle
F.
The triangle must be a right triangle
If a circle circumscribes right triangle, then its radius is
half the length of the hypotenuse
What does revolution means in circle?
one revolution = one circumference
If a wheel spins at 3 revolutions per second then how fast a point on spinning wheel travel?
3 x 2. π.r (unit of length/ second)
What is the surface area of a cube?
6a ^2
Diagonal length of a cube
a . √3
what is the other name for cuboid?
rectangular solid
What is the surface area of a cuboid?
2 (ab+ bc+ ac)
Diagonal length of a cuboid
√a^2 + b^2 + c^2
Volume of cylinder
π.r^2. h or area of circle x height
Outer surface area of cylinder w/o base
2πrh or perimeter of circle x height
Outer surface area of cylinder w/ base
2πrh + 2πr^2 or
Outer surface area w/o base + area of two circles
in a cone solid, what is a lateral height?
the hypotenuse of the triangle formed by the height and radius
Volume of cone
a third of the product of height and area of circle (π.r^2 .h )/ 3
Outer surface area of a cone w/o base
half product of circle perimeter and lateral height (πrl )
Outer surface area of a cone w/ base
Outer surface area w/o base + area of circle (πrl + πr^2)
Volume of sphere
(4/3).π. r^3
Surface area of sphere
4πr^2
For given two points (x1,y1) and (x2, y2) the distance between two points is
√(x2 - x1)^2 + (y2- y1)^2
For given two points (x1,y1) and (x2, y2) the midpoint is
x3 = (x1 + x2)/2 y3 = (y1 + y2) /2
How do you find the slope of a line?
difference of y coordinate/ difference of x coordinate
when does slope is zero happen?
when the line is horizontal. Thus
difference of y over difference of x is ZERO
when is the slope undefined?
when the line is vertical. Thus
difference of x (numerator) is zero.
If the slope equal to one, then what can we conclude?
the angle forms with the line is 45 degree
what can we say about the slopes of two parallel line?
they are constant and equal to each other
Distance between two parallel lines
k1 - k2 | / √(slope^2 + 1)
T or F. the two lines are perpendicular if and only if the product of their slopes is 1
F.
-1
if m is the slope of one line, what is the value of slope that perpendicular to that line
-1/m
The two lines a.x + b.y + c = 0 and m.x + n.y + q = 0 are perpendicular if
a.m + b.n = 0
If two lines cross each other, there is one intersected point (x,y) then at that point, the equation of line is
y(1) = y(2)
-> a.x + m = b.x + n
T or F. The distance from a point to a line is shortest when the segment from that point perpendicular to the line
T
The distance from a point (m, n) to a line ax + by+ c = 0 is given by the formula
m.a + b.n + c | / √(a^2 + b^2)
what is the expression of a parabola?
It is a quadratic expression: ax^2 + bx + c
T or F. The smaller the absolute value of a, the bigger (more open) the parabola is
T. since the value of y increase less quickly
If a is positive, the parabola open
upward
If a is negative, the parabola open
downward
at x- intercept, the parabola expression is written as
ax^2 + bx + c = 0 since y =0
at y - intercept, the parabola expression is written as
y = c since x = 0
If the discriminant is positive, then parabola has
two intercept with x-axis or two solutions of x
If the discriminant is negative, then parabola has
no intercepts with x-axis or no solutions of x
If the discriminant is zero, then parabola has
one intercept with x-axis - that is a tangent point
What is the expression of discriminant in parabola?
b^2 - 4ac
What does the vertex value represent?
the maximum or minimum value of the quadratic function
When is the vertex maximum?
when a is negative and parabola open downward
When is the vertex minimum?
when a is positive and parabola open upward
How do we calculate vertex of the parabola?
( -b/2a; c- b^2/4a )
T or F. For one line (slope = m) to be perpendicular to another, the other slope is -m
F. it must be -1 /m
T or F. When you are fitting the 3D objects into other 3-D objects, you can divide the volume of bigger object by volume of smaller object
F.
You need to know the dimension of each unit
What do you need to know when you try to determine whether an object can fit in another one?
specific dimensions: length, width and height
With given the lengths of two sides of triangle, in what conditions, the triangle has maximum area?
when the two sides meet at right angle.
Given the lengths of two sides of parallelogram, in what condition will the parallelogram have the maximum area?
when this parallelogram is a rectangular (short side meets long side at right angle)
In coordinate geometry, x-intercept is
the line intercept x-axis only and therefore y =0
In coordinate geometry, y-intercept is
the line intercept y-axis only and therefore x= 0
Area of any parallelogram
base x height
Letting R = (x,y), then R will be equidistant from (−3,−3) and (1,−3) when?
if and only if R lies on the perpendicular bisector of the line segment with endpoints (−3,−3) and (1,−3)
The equation of a straight line whose x and y intercepts are a and b, respectively, is:
x/a + y/b = 1
