Classwork 12.1 surface area
1 answer:
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Answer:
The sum of the measures of each pair of vertical angles is equal
to the sum of the measures of the intercepted arcs
Step-by-step explanation:
* Lets study this rule :Angles of Intersecting Chords Theorem.
- If two chords intersect inside a circle, then the measure of the
angle formed is one half the sum of the measure of the arcs
intercepted by the angle and its vertical angle.
Ex: Look to the attached figure:
- QS and PR are 2 chords intersect each other inside the circle
- ∠1 subtended by arc PQ , ∠3 subtended by the arc RS
- ∠2 subtended by arc RQ , ∠4 subtended by the arc PS
∴ m∠1 = [measure of arc PQ + measure of arc RS]/2
∴ m∠2 = [measure of arc QR + measure of arc PS]/2
∴ 2m∠1 = measure of arc PQ + measure of arc RS
∴ 2m∠2 = measure of arc QR + measure of arc PS
∵ m∠1 = m∠3 ⇒ vertically opposite angles
∵ m∠2 = m∠4 ⇒ vertically opposite angles
∴ m∠1 + m∠3 = 2m∠1
∴ m∠2 + m∠4 = 2m∠2
∴ m∠1 + m∠3 = measure of arc PQ + measure of arc RS
∴ m∠2 + m∠4 = measure of arc QR + measure of arc PS
* From all above
- The sum of the measures of each pair of vertical angles is equal
to the sum of the measures of the intercepted arcs
Consider the functions m and n:
i)
for x≠1.
the domain of m is all the real numbers except 1.
ii) n(x)=x-3 is a polynomial function, because x-3 is a linear polynomial.
The domain of any polynomial function is all real numbers. To determine the Range of this function, let c be a value in the range:
x-3=c, then x=3+c.
so if we want the function to produce c, we just set x=3+c, check:
n(3+c)=(3+c)-3=c.
This means that the Range of n is all real numbers.
What this means, is that the input (x) of the function n(x) can be any real number, and the output (n(x)) can also be any number.
so let n(x)=a, consider m(a):
m(a)=(a+5)/(a-1), clearly a≠1, this means n(x)=x-3≠1, that is x≠-4
This means that the domain of the composed function, m(n(x)) = say f(x),
is all Real numbers except 4.
Check: letting x be 4, would mean n(4)=4-3=1
which would mean m(n(4))=m(1)=(1+5)/(1-1)=6/0, which makes no sense.
Answer: Any function with domain R-{4}, has the same domain of (mon)(x)
i)
the domain of m is all the real numbers except 1.
ii) n(x)=x-3 is a polynomial function, because x-3 is a linear polynomial.
The domain of any polynomial function is all real numbers. To determine the Range of this function, let c be a value in the range:
x-3=c, then x=3+c.
so if we want the function to produce c, we just set x=3+c, check:
n(3+c)=(3+c)-3=c.
This means that the Range of n is all real numbers.
What this means, is that the input (x) of the function n(x) can be any real number, and the output (n(x)) can also be any number.
so let n(x)=a, consider m(a):
m(a)=(a+5)/(a-1), clearly a≠1, this means n(x)=x-3≠1, that is x≠-4
This means that the domain of the composed function, m(n(x)) = say f(x),
is all Real numbers except 4.
Check: letting x be 4, would mean n(4)=4-3=1
which would mean m(n(4))=m(1)=(1+5)/(1-1)=6/0, which makes no sense.
Answer: Any function with domain R-{4}, has the same domain of (mon)(x)