Step-by-step explanation:
1.
a perpendicular line on a chord of a circle going through the center of the circle splits the chord in half.
so, x is 2 times the missing leg of the right-angled triangle. and we are using Pythagoras :
c² = a² + b²
c being the Hypotenuse (the baseline opposite of the 90° angle).
so,
8² = 3² + (x/2)²
64 = 9 + x²/4
55 = x²/4
220 = x²
x = 14.83239697... ≈ 14.8
2.
I assume x is again the length of the whole chord (the diagram is not really clear in that regard).
the chord length using the distance from the center of the circle is
2 × sqrt(r² - d²)
r being the radius, d being the perpendicular distance from the center.
remember, the radius is half the diameter.
so, in our case
2 × sqrt ((12/2)² - 3²) = 2 × sqrt(36 - 9) = 2×sqrt(27) =
= 10.39230485... ≈ 10.4
3.
c is the supplementary angle to 76°, as the baseline of the abd triangle is the diameter of the circle, so c+76 are a half-circle.
c = 180 - 76 = 104°
the circle theorem says, the angle at the center is twice the angle at the circumference.
so,
a = 76/2 = 38°
similarly,
b = 104/2 = 52°
and
d = 180 - 38 - 52 = 90°
as it must be, as any inscribed triangle over the diameter of a circle must be right-angled.
4.
like in 3.
c = 150/2 = 75°
a + 90 + 150 = 360 (as together they are a full circle)
a = 360 - 90 - 150 = 120°
as a is the arc angle of the chord, it is also the inner angle at the center point of the triangle chord-center.
this triangle is an isoceles triangle (both invisible legs and therefore also both angles to the baseline are the same).
the sum of all angles in a triangle is 180°.
so, both leg angles of that triangle are
(180 - 120)/2 = 60/2 = 30°
b is the complementary angle to 30°, as this outside line is the tangent to the circle at the chord end (at a right angle to the triangle leg).
so,
b = 90 - 30 = 60°
5.
similar to 4.
x + 95 + 72 = 360
x = 360 - 95 - 72 = 193°
and for y let's consider the tangent secant angle theorem :
y = (arc angle of "outside" arc - arc angle of "inside"arc)/2
y = (193 - 72)/2 = 121/2 = 60.5°
