Chord, Tangent and the Circle

The Theorem: An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc


Look at the picture on the left

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  This means that the measure of arc ABC (the purple portion of the circle itself) is half the measure of angle C.

Use the theorem above to find the measure of angle formed by the intersection of the tangent that intersects chord AC.


 Measure of angle  

By the theorem, the measure of angle is half of the intercepted arc which is 170°. Therefore
x = ½ • 170 = 85°

 

What is the ?

Answer

Remember the theorem: the angle formed by the tangent and the chord is half of the mesure of the intercepted arc.
Therefore, the arc is double the angle. = 2 • 70° =140°

For (the measure of arc ABC) to equal ¾ the total measure of the circle's circumference, what must be the value of X ?

Answer

Total degree measure of of Circle's circumference = 360°
¾( 360°)=270°
By our theorem, we know that the angle formed by a tangent and a chord must equal half of the intercepted arc so
X=½(270° )=135°

Look at Circle 1 and Circlce 2 below. In only one of the two circles does a tangent interesect with a chord. Which circle is it?

Answer

Circle 1 is the only circle whose intercepted arc is half the measure of the angle between the chord and the intersecting line.