George can read 32 pages in one hour. At this rate, how many pages can he read in 90 minutes
2 answers:
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Proof of the Law of Sines
The Law of Sines states that for any triangle ABC, with sides a,b,c (see below)
a
sin A
=
b
sin B
=
c
sin C
For more see Law of Sines.
Acute triangles
Draw the altitude h from the vertex A of the triangle
From the definition of the sine function
sin B =
h
c
a n d sin C =
h
b
or
h = c sin B a n d h = b sin C
Since they are both equal to h
c sin B = b sin C
Dividing through by sinB and then sinC
c
sin C
=
b
sin B
Repeat the above, this time with the altitude drawn from point B
Using a similar method it can be shown that in this case
c
sin C
=
a
sin A
Combining (4) and (5) :
a
sin A
=
b
sin B
=
c
sin C
- Q.E.D
Obtuse Triangles
The proof above requires that we draw two altitudes of the triangle. In the case of obtuse triangles, two of the altitudes are outside the triangle, so we need a slightly different proof. It uses one interior altitude as above, but also one exterior altitude.
First the interior altitude. This is the same as the proof for acute triangles above.
Draw the altitude h from the vertex A of the triangle
sin B =
h
c
a n d sin C =
h
b
or
h = c sin B a n d h = b sin C
Since they are both equal to h
c sin B = b sin C
Dividing through by sinB and then sinC
c
sin C
=
b
sin B
Draw the second altitude h from B. This requires extending the side b:
The angles BAC and BAK are supplementary, so the sine of both are the same.
(see Supplementary angles trig identities)
Angle A is BAC, so
sin A =
h
c
or
h = c sin A
In the larger triangle CBK
sin C =
h
a
or
h = a sin C
From (6) and (7) since they are both equal to h
c sin A = a sin C
Dividing through by sinA then sinC:
a
sin A
=
c
sin C
Combining (4) and (9):
a
sin A
=
b
sin B
=
c
sin C
Answer:
B: II, IV, I, III
Step-by-step explanation:
We believe the proof <em>statement — reason</em> pairs need to be ordered as shown below
Point F is a midpoint of Line segment AB Point E is a midpoint of Line segment AC — given
Draw Line segment BE Draw Line segment FC — by Construction
Point G is the point of intersection between Line segment BE and Line segment FC — Intersecting Lines Postulate
Draw Line segment AG — by Construction
Point D is the point of intersection between Line segment AG and Line segment BC — Intersecting Lines Postulate
Point H lies on Line segment AG such that Line segment AG ≅ Line segment GH — by Construction
__
II Line segment FG is parallel to line segment BH and Line segment GE is parallel to line segment HC — Midsegment Theorem
IV Line segment GC is parallel to line segment BH and Line segment BG is parallel to line segment HC — Substitution
I BGCH is a parallelogram — Properties of a Parallelogram (opposite sides are parallel)
III Line segment BD ≅ Line segment DC — Properties of a Parallelogram (diagonals bisect each other)
__
Line segment AD is a median Definition of a Median
