Geometry Flashcards
The diagonal of a square is
root 2 times longer than the side
The side of a square is
root 2 times shorter than the diagonal
The height of an equilateral triangle is
root three times longer than half of the base
For circles, add
key radii at strategic places
Split up the shape into
manageable chunks
for a shared length find
two expressions
Draw right angled triangles using radii
Give radii length 1
Sector area =
0.5Xr^2
Arc length =
rX
Whenever one of the edges is an arc
one of your sub-areas will be a sector
When asked to find the area of a complex shape
split it up; there tends to be an easier way
Nearest point from a circle means
draw a line from the centre of the circle to the point of interest
Consider the conditions for which the equation of a circle area valid i.e
r, r^2 > 0
Exterior angle =
360/n
Interior angle =
180 – 360/n
Sine rule:
two sides + opposite angle, one side + two angles
Obtuse/acute
multiple answers
Reflect on the significance of
each piece of information given to you
We have a tangent
likely be able to use the alternate segment theorem
If two circles touch
we have a tangent, use alternate segment theorem
Given the diameter
the angle subtended on the circumference is 90 degrees
Use variables to represent appropriate unknown angles/lengths
form equations using Pythagoras or comparing lengths to find the values of these variables
Look out for similar triangles to
compare lengths
Extend or add lines where necessary
Justify your assumptions
Similar triangles
if two triangles are similar, then their ratio of width to height is the same. They are similar if have two equal angles, two sides of equal ratio and an included angle that is equal, or if three sides of equal ratios. One common occurrence is where one triangle is embedded in the other.
Circle theorem (5):
angle in semi-circle = 90 degrees,
angle in same segment is equal (2 angles subtended from the same minor arc are equal), The angles subtended by the chord at the circumference of the circle are equal.
angle between chord and tangent is equal to angle in alternate segment,
angle at circumference is half the angle at centre,
opposite angles in cyclic quadrilateral is equal to 180 degrees.
The angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points,
Angles which are in the same segment are equal, i.e. angles subtended (made) by the same arc at the circumference are equal,
The angles which are within a semicircle add up to 90°,
Opposite angles in a cyclic quadrilateral add up to 180°,
Alternate Segment Theorem, i.e. that the angle between a tangent and its chord is equal to the angle in the ‘alternate segment’.
Alternate segment theorem
chord meets tangent
When asked to find the area of a more complex shape
split it up, tends to be an easier way to do so
Whenever one of the edges is an arc
one of your sub-areas will be as sector
area of triangle with co-ordinates (0,0) (a,b) (c,d) =
0.5|ad-bc|
Point on a circle a closest to circle b:
distance between centre;s, fraction of radii to length, times that by x and y distances, +- for correct circle’s centre
triangle and circle –>
look for sectors and equal areas
sinX , cosX in triangle
look for trig length (of 1)
triangle + circle = sector
x + y = k
largest co-
ordinates on a circle and line
smallest value of a circle at a certain point
1/root 2
Always think how to get sinX cosX - if length is 1, look for triangles and trig
(sinX-cosX)^2 > 0
x^2 + y^2 or y = -(x+7)^0.5
think circle
Extreme co-ordinates
think how it changes the graphs
A(k) = A(2-k) therefore
as even function about k=1
Angle at circumference =
1/2 * angle at radius
Sin(2a) =
don’t need 2sinacosa but may be useful
Tangent to circle:
repeated root, (2y-3)^2 when equating line to circle as touches, one solution
If triangle and circle think
of sectors, equal/repeated areas
If an area has a pi, likely to involve
a sector = 0.5Xr^2
Reflect Q(x, y) in y=mx+c to find P(x,y)
PQ is perpendicular to y=mx+c, find equation of PQ and equate to y to find point of intersection M –> QM = MP
Circle inside equilateral triangle
area a is congruent to area b and c
Sphere
A = 4pir^2 V = 4/3 pi r^3
Cone
A = pi*r*(L+R) V = 1/3 * pi *h *r^2
Cylinder
A = 2pi*r*(r+h) V = pi*h*r^2
Max of x+y when xx + yy <=1
the inequality means it lies on or inside the circle
let x+y = c then y=c-x, as c increases it shifts this line upwards until it becomes a tangent to the circle and x+y = root 2
Use variables for lengths
s
Area of trapezium =
0.5h(a+b)
Point that splits P(2,3) and Q(8,-3) in ration 1:2
Difference of PQ is (6,-6) is and so we want to add (2, -2) to P or subtract (4, -4) from Q to get (4, 1)
R is a reflection of P in x axis and so if X is a point on the x axis then PX = RX,
if S is a reflection of Q in the line y=mx+c then if Y is a point on that line then QY = YS
