The circle has radius 5cm are chord PQ of the circle is of length 8cm.find the angle the chord PQ subtends at the centre of the circle and the perimeter of the minor segment
Draw the radius to the ends of the chord. You have an isosceles triangle with base of 8 and sides of 5 each.
You could use the cosine law to find the central angle.
or
drop a perpendicular from the centre to the base.
You now have a right-angled triangle with a side of 4 cm and a hypotenuse of 5 cm
you can find any angle, and from there the central angle.
let me know what you get
Draw a diagram, as always.
If the central angle is 2θ, then
tanθ = 4/5
the perimeter is the chord + the arc subtended:
8 + 5*2θ
Oops. Reiny is right. sinθ = 4/5
To find the angle that the chord PQ subtends at the center of the circle, we need to use the formula for the angle of a chord:
Angle = 2 * arcsin(length of chord / (2 * radius))
In this case, the length of the chord PQ is 8 cm and the radius of the circle is 5 cm. Plugging these values into the formula, we get:
Angle = 2 * arcsin(8 / (2 * 5))
First, we simplify the fraction inside the arcsin:
Angle = 2 * arcsin(0.8)
Now we can find the arcsin of 0.8 using a calculator. The answer we get is approximately 0.9273 radians. To convert this to degrees, we multiply by 180/π:
Angle = 0.9273 * (180/π) ≈ 53.13 degrees
So, the angle that the chord PQ subtends at the center of the circle is approximately 53.13 degrees.
To find the perimeter of the minor segment, we need to use the formula for the circumference of a circle and subtract the length of the chord. The formula for the circumference is:
Circumference = 2 * π * radius
In this case, the radius is 5 cm. Plugging this into the formula, we get:
Circumference = 2 * π * 5 = 10π cm
The length of the chord PQ is 8 cm. Subtracting this from the circumference, we get:
Perimeter of minor segment = Circumference - length of chord
= 10π - 8 ≈ 22.85 cm
So, the perimeter of the minor segment is approximately 22.85 cm.