This is one of those dredfull problems in geometry that ive got to solve using proofs. Does anyone know how to proove this using the correct theorems definitions or postulates?
1. Prove that tangents to a circle at the endpoints of a diameter are parallel. State what is given, what is to be proved, and your plan of proof. Then write a two-column proof.
God help me!
thanks for your support
-Daniel
2007-03-08 05:44:33 · 7 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
oh let me see
this proof was told to me by a friend and i guess it is just great
Because the tangents pass through the endpoints of the
diameter of the circle, the tangents are parallel if they
form congruent alternate angles. The tangents form congruent
alternate angles only if they are purpendicular to either end
point of the diameter. The tangents are purpendicular to
either endpoint of the diameter because of the theorem 9-1.
2007-03-08 09:20:11 · answer #1 · answered by emy 3 · 0⤊ 0⤋
Before going about proving that the tangents drawn at either ends of a diameter of a circle are parallel, you need to know that the radius is perpendicular to the tangent of the circle.
We are given that there is a line and two tangents to the circle to which the line is a diameter for either ends of the diameter.
We need to prove that the lines at the either end of the diameter are parallel to each other.
Two lines are parallel to each other only if a line that cuts through them makes the same alternate angles with both of them.
In our case, since the diameter is perpendicular to the tangent, the two lines are perpendicular to the diameter on either side. Each line makes an angle of 90 degrees with the diameter. Hence the alternate angles are 90 degrees.
Therefore, the two lines are parallel to each other. If you can draw a diagram, you'll be able to see it better.
2007-03-08 05:58:02 · answer #2 · answered by Shashi 2 · 0⤊ 1⤋
Given: circle O with center O and diameter AB: tanget lines CA and DB tangent at the points A and B respectively.
Prove: lines CA and DB are parallel
Step 1 is the given
Step 2 : AB is perpendicular to CA at A and AB is perpendicular to DB at B. Reason: A line tangent to a diameter is perpendicular at the point of tangency.
Step 3: CA is parallel to DB. Reason: Lines perpendicular to the same line are parallel to each other
I hope you can figure the drawing out.
2007-03-08 05:57:23 · answer #3 · answered by lizzie 3 · 1⤊ 1⤋
This is because the radius forms a right angle with the tangent:
Since both ages of the diameter are on the same line, it is perpendicular to both tangents, which leave the tangents no choise but be parallel.
2007-03-08 05:56:17 · answer #4 · answered by Amit Y 5 · 1⤊ 1⤋
two lines perpendicular to a third are parallel to each other and the two tangents are perp to diameter
2007-03-09 02:02:01 · answer #5 · answered by emy again 1 · 0⤊ 0⤋
I know your teacher won't accept this proof, but they're parallel simply because of symmetry. If one assumes the existence of parallel lines (Euclidean or Lobachevskian geometry), that's all you really need to know. .
2007-03-08 06:11:42 · answer #6 · answered by Scythian1950 7 · 1⤊ 1⤋
Both are perpendicular to this diameter.
2007-03-08 05:49:49 · answer #7 · answered by Anonymous · 0⤊ 1⤋