prove all the theorams of cirle ?? explain it
Answers
Two Radii and a chord make an isosceles triangle.
Perpendicular Chord Bisection
The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).
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
Answer:
I took pics of which proofs I was able to find
