explain all the theorems of circle
Answers
theorem 10.1: equal chords of the circle subtend equal angles at centre
theorem 10.2:the angle subtended by the chord of a circle at centre are equal then the chords are equal
theorem 10.3: the perpendicular from the centre of the circle to a chord bisects the chord
theorem 10.4:the line drawn through the centre of the chord to bisect a chord is perpendicular to the chord
theorem 10.5:there is one and only one circle passing through three given non-collinear points
theorem 10.6: equal chords of the circle are equidistant from the centre
theorem 10.7: chords equidistant from the centre of the circle are equal in length
theorem 10.8:the angles of standard by an Arc at the centre is double the angle subtended by it at any point on the remaining part of the circle
theorem 10.9: angles in the same segment of a circle are equal
theorem 10.10:a line segment joining two points substance equal angles at two other points lying on the same side on the line containing the line segment the four points lie on the circle
theorem 10.11: the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees
theorem 10.12:the sum of a pair of opposite angles of a quadrilateral is 180 degree the quadrilateral is cyclic
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Answer:
Angles Subtended on the Same Arc
Angles subtended on the same arc
Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Angle in a Semi-Circle
angle in a semi-circle
Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.
Proof
We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.
Divide the triangle in two
We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.
Two isosceles triangles
But all of these angles together must add up to 180°, since they are the angles of the original big triangle.
Therefore x + y + x + y = 180, in other words 2(x + y) = 180.
and so x + y = 90. But x + y is the size of the angle we wanted to find.
Tangents
A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).
A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.
angle with a tangent
Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.
Tangents from an external point are equal in length
Angle at the Centre
Angle at the centre
The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.
Proof
You might have to be able to prove this fact:
proof diagram 1
OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a
Similarly, ∠OXB = b
proof diagram 2
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 - 2a
Similarly, ∠BOX = 180 - 2b
Since the angles around a point add up to 360, we have that ∠AOB = 360 - ∠XOA - ∠BOX
= 360 - (180 - 2a) - (180 - 2b)
= 2a + 2b = 2(a + b) = 2 ∠AXB
Alternate Segment Theorem
Alternate segment theorem
This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
Proof
You may have to be able to prove the alternate segment theorem:
proof of alternate segment theorem
We use facts about related angles
A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90
But OAC + x = 90, so ∠OAC + x = ∠OAC + y
Hence x = y
Cyclic Quadrilaterals
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
Area of Sector and Arc Length
A sector
If the radius of the circle is r,
Area of sector = πr2 × A/360
Arc length = 2πr × A/360
In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360
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