Radius equals 1, OP is equal to 4, so the other side length must be equal to 4 because this is an isosceles triangle.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 4, O = 4, P = 1. Where A=A, B=O, C=P
Cosine O= O=
O = 86.417 Degrees
Use Law of Sines to Find Other Angles:
SO
Sin A= .998
A= A= 86.417 degrees
Sum of Triangle = 180 degrees Where x is the angle measurement of P
86.417+ x degrees = 180
172.834 + x degrees = 180
172.834 - 172.834 + x degrees = 180 – 172.834
x degrees = 7.166
P = 7.166
The angle and side length of A and O are the same. This makes perfect sense because the triangle is an isosceles and since A and O have the same side length, it makes perfect sense that they have the same angle measurement. Hence the sum of the angles of A and O has to be greater than P.
Angle Measurement:
A= 86.417 degrees
O= 86.417 degrees
P= 7.166 degrees
After strenuous thinking, I was able to arrive at the formula for OP Prime:
OP Prime=
As you increase the side lengths and keep the radius the same you limitate how big the angles can be. As you increase the side lengths and keep the raidus the same, the angles of O AND P get larger and angle P get smaller.
Let us analyze:
Radius of 1 and OP equal to 2: OP Prime = OP Prime =
Radius of 1 and OP equal to 3: OP Prime = OP Prime =
Radius of 1 and OP equal to 4: OP Prime = OP Prime =
As you can see: as you increase the side lengths and keep the radius the same OP Prime gets smaller.
Radius equals 2, OP is equal to 2, so the other side length must be equal to 2 because this is an equalateral triangle.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 2, O = 2, P = 2. Where A=A, B=O , C=P
Cosine O= O=
O = 60 Degrees
Use Law of Sines to Find Other Angles:
SO
Sin A= .866
A= A= 60 degrees
Sum of Triangle = 180 degrees Where x is the angle measurement of P
60+ x degrees = 180
120 + x degrees = 180
120-120 + x degrees = 180 – 120
x degrees = 60
P = 60
The angle and side length of A and O are the same. This makes perfect sense because the triangle is in this case an equilateral triangle and since A and O have the same side length, it makes perfect sense that they have the same angle measurement. Hence the sum of the angles of A and O has to be the same than P.
Radius equals 3, OP is equal to 2, so the other side length must be equal to 2 because this is an isosceles triangle.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 2, O =2, P = 3. Where A=A, B= O, C=P
Cosine O= O=
O = 41.409 Degrees
Use Law of Sines to Find Other Angles:
SO
Sin A= .661
A= A= 41.409 degrees
Sum of Triangle = 180 degrees Where x is the angle measurement of P
41.409+ x degrees = 180
82.818 + x degrees = 180
82.818 - 82.818 + x degrees = 180 –82.818
x degrees = 97.182
P = 97.182
The angle and side length of A and O are the same. This makes perfect sense because the triangle is an isosceles and since A and O have the same side length, it makes perfect sense that they have the same angle measurement.
Radius equals 4, OP is equal to 2, so the other side length must be equal to 2 because this is an isosceles triangle.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 2, O = 2, P = 4. Where A=A, B=O, C=P
Cosine O= O=
O = 0 Degrees
Use Law of Sines to Find Other Angles:
SO
Sin A= 0
A= A= 0 degrees
This triangle does not exist. You cannot increase the radius past 3 otherwise you would have to find the sin of 0 which is 0. Therefore, this triangle does not exist.
After strenuous thinking, I was able to arrive at the formula for OP Prime:
OP Prime=
As you increase the radius and keep the side lengths the same you limitate how big the angles can be. As you increase the radius and keep the side lengths the same, the angles of O AND P get smaller and angle P gets larger except when you exceed a raidus of four in which a triangle does not exist..
Let us analyze:
Radius of 2 and OP equal to 2: OP Prime = OP Prime = 2
Radius of 3 and OP equal to 2: OP Prime = OP Prime = 4.5
Radius of 4 and OP equal to 2: OP Prime = OP Prime = 8
As you can see: as you increase the radius and keep the side lengths the same OP Prime gets larger.
Comments: Is it Consistent
My formula to model the above was the same as my earlier general statement, but my general statement is different. The first one as you increase the side lengths and keep the radius the same OP Prime gets smaller and my second was as you increase the radius and keep the side lengths the same OP Prime gets larger, so changing the radius or side lengths affect what OP Prime is going to be. It would be either smaller or bigger than side P.
Using technology and using the program Geogeobra: I was able to investigate other values of r and OP.
Radius equals 2; OP is equal to 1.75, so the other side length must be equal to 1.75 because this is an isosceles triangle.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 1.75, O = 1.75, P = 2. Where A=A, B=O, C=P
This triangle does not exist. You cannot increase the radius past otherwise you would have to find the sin of 0 which is 0. Therefore, this triangle does not exist.
Radius equals 2, OP is equal to 2.9, so the other side length must be equal to 2.9 because this is an isosceles triangle. Where my radius is my x coordinate and my y coordinate is Side O.
Use Law of Cosines To Find Angle:
Cos O = ------ Where A= 4, O = 4, P = 1. Where A=A, B=O, C=P
Cosine O= O=
O = 61.576 Degrees
Sum of Triangle = 180 degrees Where x is the angle measurement of P
61.576+ x degrees = 180
123.153 + x degrees = 180
123.153 - 123.153 + x degrees = 180 – 123.153
x degrees = 56.848
P = 56.848
The angle and side length of A and O are the same. This makes perfect sense because the triangle is an isosceles and since A and O have the same side length, it makes perfect sense that they have the same angle measurement. Hence the sum of the angles of A and O has to be greater than P.
Angle Measurement:
A= 61.576 Degrees
O= 61.576 Degrees
P= 56.848 degrees
After strenuous thinking, I was able to arrive at the formula for OP Prime:
OP Prime=
As you increase the radius and keep the side lengths the same you limitate how big the angles can be. As you increase the radius and keep the side lengths the same, the angles of O AND P get smaller and angle P gets larger except when you exceed a radius of four in which a triangle does not exist..
Let us analyze:
Radius of 2 and OP equal to 2.9: OP Prime = OP Prime = 1.379
As you can see: as you increase the radius and keep the side lengths the same OP Prime gets larger.
Comments: Is it Consistent
My formula to model the above was the same as my earlier general statement, but my general statement is different. The first one as you increase the side lengths and keep the radius the same OP Prime gets smaller and my second was as you increase the radius and keep the side lengths the same OP Prime gets larger, so changing the radius or side lengths affect what OP Prime is going to be. It would be either smaller or bigger than side P.
Scopes and Limitations:
The different scopes and limitations is that the radius can never be bigger than the product of the other side lengths because you would have to end taking the cosine inverse of a number over one which never works and results in no triangle. The second thing is this formula only applies to when you are able to form a triangle so this formula does not hold true every time. So before you do anything you have to find your angles measurements of the triangle and make sure everything is in order to proceed form there. I arrived at my general statement by deriving my angles and comparing them. Also, I was able to work with my fellow classmates so I was bale to produce a general statement with everyone’s brains combined. as you increase the side lengths and keep the radius the same OP Prime gets smaller and my second was as you increase the radius and keep the side lengths the same OP Prime gets larger, so changing the radius or side lengths affect what OP Prime is going to be. It would be either smaller or bigger than side P.