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 diagram. The angle θ subtended by PQ can be found using
sin(θ/2) = 4/5
p = PQ + rθ = 8+5θ
Answer
To find the angle that the chord PQ subtends at the center of the circle, we can use the formula for the angle subtended by a chord:
θ = 2 * arcsin(d / (2 * r))
where θ is the angle subtended by the chord, d is the length of the chord, and r is the radius of the circle.
In this case, the length of chord PQ is 8 cm and the radius of the circle is 5 cm. Plugging these values into the formula, we get:
θ = 2 * arcsin(8 / (2 * 5))
= 2 * arcsin(0.8)
≈ 112.38°
Therefore, the angle that the chord PQ subtends at the center of the circle is approximately 112.38 degrees.
Now, let's find the perimeter of the minor segment. A minor segment is the region bounded by a chord and the arc that it intercepts. To find the perimeter of the minor segment, we need to calculate the length of the arc and the length of the two line segments that connect the endpoints of the chord to the arc.
The length of the arc can be found using the formula:
arc length = θ/360° * 2πr
where θ is the angle subtended by the arc and r is the radius of the circle.
In this case, we already calculated the angle θ to be approximately 112.38 degrees and the radius of the circle is 5 cm. Plugging these values into the formula, we get:
arc length = 112.38/360 * 2π * 5
= 0.312π cm
Next, we can find the length of the two line segments that connect the endpoints of the chord to the arc. These line segments form a triangle with the chord. Since the chord and the line segments are radii of the circle, they are congruent.
The length of these line segments can be found using the Pythagorean theorem:
l^2 = (r^2) - (d/2)^2
where l is the length of the line segment, r is the radius of the circle, and d is the length of the chord.
In this case, the length of chord PQ is 8 cm and the radius of the circle is 5 cm. Plugging these values into the formula, we get:
l^2 = (5^2) - (8/2)^2
= 25 - 16
= 9
l = √9
= 3 cm
Since the line segments are congruent, the total length of the line segments is 2 * 3 = 6 cm.
Finally, to find the perimeter of the minor segment, we sum the length of the arc and the length of the line segments:
perimeter = arc length + 2 * length of line segments
= 0.312π + 2 * 6
≈ 0.312π + 12.57
≈ 15.23 cm
Therefore, the perimeter of the minor segment is approximately 15.23 cm.