Probability and sector areaProbability in geometry can be expressed in terms of area.
𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑒𝑣𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑒𝑣𝑒𝑛𝑡𝑠
For example:
The probability of randomly throwing a dart and hitting the shaded area is four favorable events
4
21
out of 25 total possible events- . Its complement (probability of NOT hitting shaded) is if we
25
25
decide to count the un-shaded boxes. If too many to count, we can also find this complement
with the idea that the total probability will not go over 1 or 100%. So by subtracting the 4/25
from one we find 21/25.
The probability of randomly throwing a dart and hitting the shaded area is the shaded area over the
total area. To find the shaded area we need to find the total area and subtract the un-shaded area
as we know how to find the area of a circle. r=6 so area of circle is 36π and the total area is 144. So
144−36π
36π
the probability of hitting the shaded area is
and its complement is
.
144
144
Most students had difficulty in understanding how to find a shaded area if there was not a formula
to find it directly. The idea is that you can find the total area and the un-shaded area through
formulas. This means you need to subtract (or take away) the known area from the total area. Or
in other words- the idea of square less the circle gives you the shaded area.
r=1
Again, the idea of hitting the shaded area (of which the complement is Not shaded or hitting the
circles) we need to find the area of the shaded area. We are given the radius of one circle is one
therefore the area of one circle is 𝞹 (pi times 1 squared). All 16 circles have and area equal to 16𝞹.
The area of the square is 8x8 =64 units (The diameter of each circle is 2 and we have 4 circles wide by
64−16𝜋
64−16𝜋
4 circles tall). So the probability of hitting the shaded area is 64 and the complement is 1− 64
or
16𝜋
64
𝜋
4
𝑤ℎ𝑖𝑐ℎ 𝑟𝑒𝑑𝑢𝑐𝑒𝑠 𝑡𝑜 .
To find the area of a sector, we want the percentage of the area inside the sector as it relates to the
area of the entire circle. This will not be a given formula. We should remember how to find area of a
circle- 16𝞹 in this case (4²=16). The percentage of the total is from the idea that there is 360⁰ total
25
𝟏𝟎
and we know we want 25⁰ in this case. So 360 × 16𝞹 = 𝟗 𝝅
We then investigated several new definitions. Of particular interest was the angle
bisector. If we take an angle bisector of each of the three vertices of a triangles,
we find the INCENTER. The incenter is where the point of concurrency occurs of
the three angle bisectors.
If a definition is not listed below, check your notes or Holts website/faculty/ my
name/ and Helpful resources “definitions study guide”
We also investigated perpendicular bisectors. These lines cut the sides of the
triangles in halves (bisect) at right angles. The point of this concurrency is called
the CIRCUMCENTER.
We also investigated the altitude of a triangle which is the further distance from
any base of a triangle. Each triangle vertex is the highest point and the shortest
distance from this point to the base is a perpendicular line. This point of
concurrency is called the ORTHOCENTER.
The last point of concurrency is called the CENTROID. If you remember, we hung
a bunch of them from the ceiling. The point of concurrency comes from the three
medians of a triangle. The median is from the vertex to the midpoint of the
opposite side. The midpoint, of course, is the middle point of the opposite side.
The Centroid is also the center of gravity for a triangle.
Altitude of triangle- a segment from a vertex that is perpendicular to the opposite side or to the line containing the opposite side
(that is the book for you). My definition- The height of the triangle from its base (a measurable segment); this segment is the
shortest distance so it is perpendicular to the base. / there are three for each triangle
Median of a triangle- a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. There are three
for each triangle.
Perpendicular bisector- A perpendicular line that splits a segment into two equal points; goes through the midpoint.
Midpoint- the one and only point in the middle of a segment.
Angle bisector- a ray or segment that splits an angle into two equal angles.
Concurrent- two or more lines or segments that intersect at a single point
Point of concurrency- the point where two or more lines meet together (and yes, this is a vertex too).
Incenter of triangle- three angle bisectors meet at point of concurrency- a circle can be inscribed (touches all the sides) from this
point.
Circumcenter of a triangle- three perpendicular bisectors meet a point of concurrency. A circle can be drawn that circumscribes
(touches all the vertices) the triangle at this point.
Orthocenter- the three altitudes of a triangle meet at this point of concurrency
Centroid- the three medians of a triangle meet at this point of concurrency. This is the center of gravity for a triangle. The centroid
is 2/3rds of the distance from each vertex to the midpoint of the opposite side.
1.
2.
3.
4.
5.
6.
7.
: AAA is NOT a way to prove congruence (similar shape dilated)
: SSS- If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
: SAS- If two sides and the included angle of one triangle are congruent to two sides and the included triangle, then the two
triangles are congruent.
: SSA- is NOT a way to prove congruence (makes an ____ of me if I do)
: ASA- If two angles and the included side of one triangle are congruent to two angles and the included side of a second
triangle, then the two triangles are congruent.
: AAS- if two angles and a non-included side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent.
: CPCTC- Congruent Parts of Congruent Triangles are Congruent
We also have congruency statements like △DEF≅△JKL where each letter corresponds to each other.
Vertical angles- when two lines cross, the angles across from each other are congruent and their measures are equal.
Linear angles- are two angles that lie on a line and are supplementary and adjacent.
Supplementary angles- Two angles that add up to 180 degrees
Complementary angles- Two angles that add up to 90 degrees
Corresponding angles are
congruent if the transversal
crosses two parallel lines.
Alternate Interior Angles
are congruent if the
transversal crosses two
parallel lines.
Alternate Exterior Angles
are congruent if the
transversal crosses two
parallel lines.
Consecutive interior Angles
are supplementary if the
transversal crosses two
parallel lines.
Conjecture 24: For any parallelogram, the measures of the diagonal angles are equal.
Conjecture 25: For any parallelogram, the measures of the consecutive angles are supplementary.
Angle 1 is ___congruent___ to angle 5 because_corresponding angles
Angle 4 is _supplementary__to angle 2 because_linear angles
Angle 8 is ___ congruent __ to angle 5 because _vertical angles
Angle 15 is __ congruent ____ to angle 10 because_alt. exterior angles
Angle 14 is__ congruent __ to angle 11 because_alt. interior angles
Angle 7 is ___congruent __ to angle 11 because corresponding angles
Angle 14 is _ supplementary _to angle 9 because _consecutive interior
angles
Cheat sheet for intersections, finding slope, and finding the equation given slope and point\
•To find the intersection the steps are:
5
54
1. Set the equations equal. For example: given two equations 𝑦 = 8 𝑥 + 8 𝑎𝑛𝑑 𝑦 = 9𝑥 − 10
5
𝑥
8
54
+ 8 = 9𝑥 − 10 (just substituting one equation in for y)
2. Solve for x: Algebra 1a
3. Substitute your solution in one of the equations to find the y
𝑦 = 9𝑥 − 10 𝑠𝑜 𝑦 = 9(2) − 10 𝑤ℎ𝑖𝑐ℎ 𝑓𝑖𝑛𝑑𝑠 𝑦 = 8 𝑠𝑜 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 (2,8)
•To find the slope:
1. Use the formula m =
𝑦2 −𝑦1
𝑥2 −𝑥1
2. Simply plug into the formula and find the slope m
Suppose we have the points (2 , 2)and (−2,
3 7
11
).
2
I chose to do it as such:
11 7
−
𝑚 = 2 2 𝑤ℎ𝑖𝑐ℎ 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑠 𝑡𝑜
3
−2 −
2
4
2 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑢𝑔𝑙𝑦 𝑠𝑜 4 ÷ −7 𝑖𝑠 4 × −2 = −8 = −4
−7
2
2
2
7
14
7
2
Label one (𝑥1 , 𝑦1 ) 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 (𝑥2 , 𝑦2 )𝑎𝑛𝑑 𝑝𝑙𝑢𝑔 𝑖𝑛.
𝑛
𝑑
Or use your calculator’s key function to simplify.
•To find an equation given two points first find the slope using the above steps and then proceed to the next section.
•To find an equation given a point and slope:
1. Use the formula 𝑦2 − 𝑦1 = 𝑚(𝑥2 − 𝑥1 ) and substitute your points into 𝑥1 𝑎𝑛𝑑 𝑦1 and your slope into m
4
11
Using a given slope of − 7 𝑎𝑛𝑑 𝑝𝑜𝑖𝑛𝑡 ( 2 , −2) We can substitute into the equation as such𝑦 − −2 =
−4
11
(𝑥 −
)
7
2
2. Manipulate into slope-intercept form (y = m x + b) for easy graphing.
Some circle, tangent, secant facts/definitions
Finding the length of the arc or the degree given the arc length and radius:
Finding arc length is just like find the area of a sector. Instead of finding the percent of the total area, we are finding the
percent of the circumference.
What if the arc length and radius is given and we are looking for the central angle (an angle from the center)?
To find a missing length given a tangent line and radius.