What is the SAS Theorem?
The SAS or Side-Angle-Side Theorem is a theorem to determine congruence. Congruence is when two figures have the same shape an size. The SAS Theorem establishes that if two sides of a triangle and the angle in between them is the same in two different triangles, then the triangles are congruent (the same). For example, if you have a triangle ABC and a triangle XYZ. You know that side A and side X are equal in length and that side B and Y are equal in length AND that the angle between side A and B equals the angle between X and Y, then you can confidently say that the triangles are congruent (the same).
Knowing the origins and insertions of the pectoralis major muscle, what are the actions does the pectoralis major initiates?
Knowing the origins and insertions of a muscle are very helpful in getting an idea of what bodily actions the muscle can initiate. Often times with muscles in the body, the muscle will contract towards the origin; therefore, the contraction causes a movement in a pathway similar to the path from the insertion to the origin. The pectoralis major originates on the anterior surface of the medial half of the clavicle, the lateral part of the sternum and costal cartilage 1-6. The insertion of the pectoralis major is the intertubercular groove on the humerus. In simpler terms, the pectoralis major fans across your chest from the center to the outside, and it inserts on the outside of your upper arm, right before it turns into your shoulder. These facts about the pectoralis major are important in understanding the actions that it initiates on the body. The actions that the pectoralis major initiates are adduction, medial rotation and flexion of the humerus. The definition of adduction is the movement towards the midline of your body. Adduction of the humerus can be physically done if you extend your arms to each side (away from your body), and then lower your arms to your sides, the lowering of your arms to your sides would be adduction. This can be related to the origins and insertions because once your arm is extended away from your body, the pectoralis major will be stretched out. Therefore, upon contraction it will lower your arm down to your side since the insertion point is on your upper arm. The medial rotation of the humerus means that your arm rotates in towards your body. This connects with the origin and insertions of the pectoralis major because the intertubercular groove of the humerus is on the outer side of your arm. Therefore, when the muscle contracts, it will rotate your arm with it. The flexion of the humerus is the action of extending your arm forward and away from your body when the starting point is your arm relaxed at your side. The pectoralis major muscle helps with this movement.
If f(x) = (5*x + 10) / (15 - 3*x), what is f(10) - f(4)?
F(x) is a mathematical term to say "the function" of x. In other words, f(x) is providing an equation where the variable (or thing) that is changing in the equation is x. Therefore, if you are given the equation: f(x) = (5*x + 10) / (15- 3*x) And you are told to find out what f(10) - f(4) is then 10 and 4 are your two different values for x that will change the output (or value) of the equation. To solve this problem you want to split up "f(10) - f(4)" into two parts. You want to first solve for the value of f(10) and then solve for the value of f(4). To solve for f(10): wherever there is an "x" in your f(x) equation you want to plug in the value 10. f(x) = (5*x + 10) / (15- 3*x) -------> f(10) = (5*10 + 10) / (15 - 3*10) Next, you want to simplify the numerator and denominator. The numerator is (5*10 + 10). Therefore, you want to first multiply 5*10 which is 50, and then do (50+10) which is 60. The denominator is (15 - 3*10). Therefore, you want to first multiply 3*10 which is 30, and then do (15-30) which is -15. Now that the numerator and denominator are simplified, the f(10) equation reads: f(10) = (60) / (-15) Since, a negative number is the same as multiplying a positive number by -1, you can rewrite the f(10) equation as follows: f(10) = (-1) * (60) / (15) You can then divide 60 by 15 which is 4. Then multiply 4 * -1, which is -4. Therefore, f(10) = -4. You can follow the same steps as above to get the value of f(4). The value of f(4) = 10. You can then insert the values of f(10) and f(4) into the original question: f(10) - f(4) = ? ----------> (-4) - (10) = -14