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March 10th, 2016, 02:29 PM   #1
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Circle theorems and their converse

I kind of don't understand what the converse of these three theorems are:

1. The angle in a semi-circle is a right angle.
2. The perpendicular from the centre of a circle to a chord bisects the chord.
3. The tangent to a circle at a point is perpendicular to the radius through that point.

Can anyone specificy? I am thinking 2 is that the perpendicular bisector of the chord goes to the centre of the circle but it sounds awful to be honest.

I am mostly focused on identifying the converse of 1. I have a question which asks the following:
A circle passes through the point A (3,2), B(5,6) and C (11,3). Calculate the lengths. Hence show that AC is a diameter of this circle. State which theorems you have used, and in each case whether you have used the theorem or its converse.

I have found lengths. And now the hence bit is something I am rather stuck on because its definitely theorem 1 but I cannot explain to myself why it wouldn't be its converse since I don't understand the converse. Even without it asking me to state it, the fact that AC is a diameter is not actually stated within the theorem but rather it is the result.

Can anyone give me the converse of the three theorems?
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March 10th, 2016, 07:40 PM   #2
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Originally Posted by Tangeton View Post
I kind of don't understand what the converse of these three theorems are:

1. The angle in a semi-circle is a right angle.
2. The perpendicular from the centre of a circle to a chord bisects the chord.
3. The tangent to a circle at a point is perpendicular to the radius through that point.

Can anyone specificy? I am thinking 2 is that the perpendicular bisector of the chord goes to the centre of the circle but it sounds awful to be honest.

I am mostly focused on identifying the converse of 1. I have a question which asks the following:
A circle passes through the point A (3,2), B(5,6) and C (11,3). Calculate the lengths. Hence show that AC is a diameter of this circle. State which theorems you have used, and in each case whether you have used the theorem or its converse.

I have found lengths. And now the hence bit is something I am rather stuck on because its definitely theorem 1 but I cannot explain to myself why it wouldn't be its converse since I don't understand the converse. Even without it asking me to state it, the fact that AC is a diameter is not actually stated within the theorem but rather it is the result.

Can anyone give me the converse of the three theorems?
1. It starts from the fact that it's a semicircle, and ends with the result that there's a right angle. So the converse starts from the fact that there's a right angle, and ends with the fact that it's a semicircle, which directly implies that the chord is a diameter.
2. You have to be very careful with this one, because there are three factors in play, and three 2 premises + 1 conclusion combination work:
a) Line runs from centre to chord + Perpendicular to chord = Bisects chord
b) Perpendicular to chord + Bisects chord = Line runs from centre to chord
c) Bisects chord + Runs from centre to chord = Perpendicular to chord
a) is the one you have, b) is stated as 'perpendicular bisector of chord passes through centre', and c) is stated as 'angle from centre to mid-point of chord bisects chord'.
3. If a normal is perpendicular to the centre at the point where it intersects the circumference of the circle, then it's a tangent.
Thanks from Tangeton
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