A chord subtends a central angle of 60deg. in a circle of rad. 30cm.?

Question

Another chord subtends a central angle of 120deg. in a circle of rad. 15cm. which chord is longer? Show calculations.

Answer 1

A chord subtends a central angle of 60deg. in a circle of rad. 30cm.

This is an equilateral triangle.
Thus chord length = 30 cm

Another chord subtends a central angle of 120deg. in a circle of rad. 15cm.

Well the diameter of this circle is 30 cm (2 * radius) so the length of this chord < 30 cm

Thus the first chord is longer

(If you want the length of the 2nd chord, Bisect angle to meet the chord. This is a right angled triandle with a radius as the hypotenuse and half the chord opposite the half of 120° ie 60°

Thus sin 60° = (chordlength/2)/Radius)
= chord ength/(2*15)
= chordlength/30
This chordlength = 30sin60°
= 30*√(3)/2
=15*√(3)
≈ 25.98 cm
< 30 cm ... as stated above)

Answer 2

1> two radii and the chord form a triangle. the two equal radii and the central angle 60* suggest the triangle is equilateral. So chord = 30cm in length
2> length of chord = sqrt(15^2 + 15^2 - 2*15*15*cos(120*))
=sqrt(15^2*3) = 15 * 1.732 < 30

So first chord is longer

Answer 3

1) Let the chord subtends an angle 2A the center, which is the central angle. 2) The line perpendicular from center to the chord bisects the chord. As well it bisects the central angle. So sin(A) = opp/hyp = (1/2)chord/radius = 15/20 = 3/4 This gives A = 48.6° (nearly) Hence 2A = 97.2° (nearly)

Answer 4

Arclength= Degree subtended in radians * radius

Degree/180 = radian/pi

60 deg = pi / 3

120 deg= 2 (pi / 3)

arc1= 30 * pi/3 = 10 * pi

arc2=2 (pi/3)*15 = 10 * pi

arc1=arc2

Both arcs are of same length ; that is

= 10 * pi [centimeters]

but we need chord length comparison

Employ sine theorem

chord1/sin(60)=30/sin(60)

chord1=30 cms

chord2/sin(120)=chord2/sin(60) = 15/sin(30)

chord2=15*sin(60)/sin(30)

=15*2*sqrt(3)/2

chord2=15*sqrt(3)

3<4
sqrt(3)
sqrt(3)<2

multiply both sides by 15

15*sqrt(3)<15*2
15*sqrt(3)<30

chord2
BUT please note that , in first case what's formed is an equilateral triangle ( all angles 60 degrees), all sides equal; that is 30 cms

in the second case, ,it's an isosceles with (120,30,30) angles inside the triangle.

Answer 5

Both chords form an isosceles triangle with the radii. Using the law of cosines for triangles. The chord length C^2 = r^2+r^2-2r^2cos(T).

C =r* √[2(1-cos(T)]

You can now calculate both chord lengths.

Answer 6

Bisecting the chord also bisects the angle:
d1 = 2*30sin(60/2) = 30 cm
d2 = 2*15sin(120/2) = 25.98 cm
d1 > d2