unit 8.1  property of tangents in a circle
what do I need to know?
 What is a radius?
 What is a diameter?
 Be able to identify and differentiate the radii (plural of radius) from the diameter.
 Perpendicular lines form right angles. That is, 90 degree angles on each side of the point of intersection.
 What is a diameter?
 Be able to identify and differentiate the radii (plural of radius) from the diameter.
 Perpendicular lines form right angles. That is, 90 degree angles on each side of the point of intersection.
more information
In this part of Unit 8, we will be using the property of tangents in circles to find either AN ANGLE or the LENGTH of a SIDE LEG or the HYPOTENUSE of right triangles. Before we do, let's review the basic terminology of circle geometry:
the Pythagorean theorem
Remember that we have used, and therefore will use, the Pythagorean Theorem for RIGHT TRIANGLES ONLY!
what is a tangent?
In geometry, we define a TANGENT as a line that touches the circle exactly in one spot, never entering the circle's interior.
what is the point of tangency?
The point where the tangent touches the curve is the point of tangency.
When a radius of a circle is drawn to a POINT OF TANGENCY (from the center, of the circle, of course), that radius is perpendicular to the tangent line containing that point of tangency. This means that for any tangent line, there exists a perpendicular radius.
using the property of tangents in circles 
finding the missing length
Look at the previous picture (above). You will notice that the radius and the tangent are perpendicular to each other. That is, the radius and the tangent, at the point of tangency, form a RIGHT ANGLE (90 degrees). This means that we can use the PYTHAGOREAN THEOREM to find the lengths of the side legs or the hypotenuse of the right triangle formed once we draw a line joining the center of the circle and the tangent. Let's look at some examples:
using the property of tangents in circles 
finding the missing angle
In order to find the missing angle, remember that:
 The sum of the 3 angles in all rectangles is ALWAYS equal to 180 degrees.
 Even if not directly written, the angle at the POINT OF TANGENCY is ALWAYS 90 DEGREES.
 This means that, in order to find the missing angle, you will always know 2 out of the three angle
To find the missing angle:
 Add the two angles you have (Remember: the angle at the point of tangency is always a right angle of 90 degrees).
 Subtract the above sum from 180. This will be the missing angle.
Look at the following examples:
 The sum of the 3 angles in all rectangles is ALWAYS equal to 180 degrees.
 Even if not directly written, the angle at the POINT OF TANGENCY is ALWAYS 90 DEGREES.
 This means that, in order to find the missing angle, you will always know 2 out of the three angle
To find the missing angle:
 Add the two angles you have (Remember: the angle at the point of tangency is always a right angle of 90 degrees).
 Subtract the above sum from 180. This will be the missing angle.
Look at the following examples:
 The point of tangency is at C.
 The angle at C, therefore, is 90 degrees.
 Next, we must add the two known angles: 42 + 90 = 132 degrees.
 Now subtract 132 degrees from 180 degrees: 180  132 = 68 degrees.
 Therefore X = 68 degrees.
 The angle at C, therefore, is 90 degrees.
 Next, we must add the two known angles: 42 + 90 = 132 degrees.
 Now subtract 132 degrees from 180 degrees: 180  132 = 68 degrees.
 Therefore X = 68 degrees.
 The point of tangency is at J.
 The angle at J, therefore, is 90 degrees.
 Now, we must add the two known angles: 14 + 90 = 104 degrees.
 Now subtract 104 degrees from 180 degrees: 180  104 = 76 degrees.
 Therefore X = 76 degrees.
 The angle at J, therefore, is 90 degrees.
 Now, we must add the two known angles: 14 + 90 = 104 degrees.
 Now subtract 104 degrees from 180 degrees: 180  104 = 76 degrees.
 Therefore X = 76 degrees.
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