Question

1. Find the value of x to the nearest tenth.

Answer 1

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°