Explanation:
1.
a. true
b. false (the arc angles are relative to the center of the circle)
c. true (as the central angle of a circle is twice any inscribed angle subtended by the same arc)
d. true (but note that the arc angle of YE is again measured via the center of the circle and twice as large as the angle S)
2.
EGS is an isoceles triangle (that means both legs are equally long).
remember the extended Pythagoras for general triangles :
c² = a² + b² - 2ab×cos(C)
c being the side opposite of the angle C.
let's say c is our long baseline, since we have is opposite angle 124°, and a = b = GE = GS, we have
3.8² = a² + a² - 2aa×cos(124) = 2a² - 2a²×cos(124) =
= 2a²(1 - cos(124))
2a² = 3.8²/(1 - cos(124))
a² = 3.8²/(2(1 - cos(124)))
a = GS = sqrt(3.8²/(2(1 - cos(124)))) =
= 2.151883096... ≈ 2.2 cm
3.
circumference
16pi = 2pi×r
r = 8
area = pi×r² = pi×8² = 64pi
yes, the statement is true.
4.
a half-circle has 180°.
the arc angel AD is a full half-circle (because AD is a diameter), so it is 180°.
the angle ACB is the arc angle AB = 40°. so, the arc angle BD must be the rest of the 180° = 180 - 40 = 140°.
5.
diameter = 50 ft.
so, the radius or distance from the center to the circle arc is 50/2 = 25 ft.
the length of a chord is
c = 2×sqrt(r² - d²)
with r being the radius, d being the distance of the chord from the center.
40 = 2×sqrt(25² - d²)
20 = sqrt(625 - d²)
400 = 625 - d²
-225 = -d²
225 = d²
d = 15 cm
so, the distance of the chord from the center is 15 cm.
6.
the arc angles RE and TS are equal. that means the arc lengths of RE and TS are equal.
and that means the arc lengths of RS and ET are equal. the chords between 2 points on a circle must have the same length, when the arc lengths between these 2 points are the same.
7.
they are equally long.
as we saw in 5., the length of a chord depends on the radius of the circle and the distance of the chord from the center.
since both chords are in the same circle (with the same radius), and they both have the same distance from the center, they can only be equally long, as there is no other element to name them different.
8.
the angle NPE is half of the angle NAE (= the arc angle NE).
the arc angle NE = 63×2 = 126°
the angle PET is half of the angle PAT (= the arc angle PT).
the arc angle PT = 19×2 = 38°
the arcs NP and ET cover the rest of the whole circle circumference (360°).
so, arc angle NP + arc angle ET = 360 - 38 - 126 = 196°
but arc angles NP and ET must be equal (as per the problem definition).
therefore, 2 equal arc angles = 196°
1 arc angle = NP = 196/2 = 98°