A circle has a chord that goes from #( pi)/3 # to #(5 pi) / 12 # radians on the circle. If the area of the circle is #7 pi #, what is the length of the chord?
1 Answer
≈ 2.02
Explanation:
I will attempt to explain the solution with no recourse to diagrams , although I recommend that you make sketches as we go.
If you draw the radii from the centre of the circle to both ends of the chord , then you will have a triangle.
Now the angle subtended at the centre by the chord will be.
# (5pi)/12 - pi/3 = pi/4 # The radius of the circle may be found , since we are given it's area, by using the following.
# pir^2 = 7pi →r^2 =( 7 cancel(pi))/cancel(pi) =7 → r = sqrt7# The triangle now has 2 sides (radii) =
#sqrt7 #
with an angle between them of# pi/4 #
and a third side - the chord.To find the length of the chord , from the triangle, we need to use the
#color(blue)" cosine rule "#
# a^2 = b^2 + c^2 - (2bc costheta ) # For this question a , is the chord , b and c , the radii and
# theta = pi/4# so
# a^2 = (sqrt7)^2+(sqrt7)^2 -(2xxsqrt7xxsqrt7xxcos(pi/4))#
# = 7 + 7 - (14cos(pi/4)) = 14 - (9.899) = 4.101 # now
# a^2 =4.101 rArr a " ( length of chord)" = sqrt4.101 ≈ 2.02#
