The circle has radius 5cm are chord PQ of the circle is of length 8cm.find the angle of the chord PQ subtends at the centre of the circle and the perimeter of the minor segment?
make your sketch
join the centre to the ends of the chord.
You now have an isosceles triangle with base of 8 and sides of 5
Let the centre be C, by the cosine law:
8^2 = 5^2 + 5^2 - 2(5)(5)cosC
solve for C
The rest is easy
or, you have two right triangles with leg 4 and hypotenuse 5.
If the central angle subtended by the chord is 2x, then
sin x = 4/5
(From above, C = 2x)
We had the same question yesterday, but I can't find it.
The search does not work as nice as it used to for me.
To find the angle that chord PQ subtends at the center of the circle, we can use the theorem that states an angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the circumference.
Step by step solution to finding the angle of the chord PQ subtending at the center of the circle:
1. Draw the diagram: Draw a circle with center O and radius 5cm. Place points P and Q on the circumference of the circle such that the chord PQ has a length of 8cm.
2. Find the midpoint of chord PQ: Mark the midpoint of chord PQ, let's call it M. The midpoint divides the chord into two equal parts, each of length 4cm.
3. Connect point M to the center O: Draw a line segment MO connecting the midpoint M of the chord PQ to the center of the circle O.
4. Calculate angle MOQ: Since MO is a radius of the circle, we know that angle MOQ is a right angle (90 degrees).
5. Calculate angle MQP: The length of chord PQ is 8cm, and it is divided into two parts by the midpoint M. Each part has a length of 4cm. So we have a right-angled triangle MPQ with sides 4cm and 8cm. To find angle MQP, we can use trigonometric functions. In this case, we can use the inverse tangent function (tan^(-1)). Let's calculate this angle.
tan(MQP) = (opposite side) / (adjacent side) = (MP / OQ) = (4 / 5)
MQP = tan^(-1)(4/5) = 38.66 degrees (approx)
6. Calculate angle PQO: Since angle MOQ is a right angle (90 degrees) and angle MQP is 38.66 degrees, we can use the theorem mentioned earlier: angle PQO is twice angle MQP.
angle PQO = 2 * angle MQP = 2 * 38.66 = 77.32 degrees (approx)
Therefore, the angle of the chord PQ subtends at the center of the circle is approximately 77.32 degrees.
To find the perimeter of the minor segment, we need the length of the arc formed by the minor segment. We can calculate this using the formula for the circumference of a circle.
7. Calculate the circumference of the circle: The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. In this case, r = 5cm.
C = 2π * 5 = 10π cm
8. Calculate the angle subtended by the chord PQ at the center of the circle: We've already calculated this in step 6. The angle is approximately 77.32 degrees.
9. Calculate the length of the arc formed by the minor segment: The arc length can be calculated using the formula S = (θ/360) * C, where S is the arc length, θ is the angle in degrees, and C is the circumference.
S = (77.32/360) * (10π) ≈ 6.82 cm
10. Calculate the perimeter of the minor segment: The perimeter of the minor segment consists of the length of the chord PQ and the length of the arc formed by the minor segment.
P = chord length + arc length = 8 + 6.82 ≈ 14.82 cm
Therefore, the perimeter of the minor segment is approximately 14.82 cm.