Circles: The Angle formed by a Chord and A Tangent, Intercepted Arc. Examples, Pictures, Interactive Demonstration and Practice Problems

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$$ \angle X = \class{data-anglex-0}{35.92} \\ \overparen{\rm ABC} = \class{data-anglex-1}{35.92} \\ \frac 1 2 \overparen{\rm ABC} = \class{data-anglex-0}{35.92} $$

$$ \angle Y = \class{data-angley-0}{35.92} \\ \overparen{\rm ADC} = \class{data-angley-1}{35.92} \\ \frac 1 2 \overparen{\rm ADC} = \class{data-angley-0}{35.92} $$

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Practice Problems

Problem 1

Chord $$ AC $$ intercepts a tangent tangent at point C. If the measure of $$ \overparen{abc} = 70^{\circ} $$, what is x ?

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

By the theorem, the measure of angle is half of the intercepted arc which is $$70 ^{\circ}$$ .

Picture of angle formed by tangent and chord

Therefore $$ x = \frac{1}{2} \cdot 70 = 35^{\circ} $$ .

Problem 2

Chord $$ AC $$ intercepts a tangent tangent at point C. If the measure of $$ \overparen{abc} = 110^{\circ} $$, what is x ?

What is the $$ m\overparen{ABC} $$(the measure of arc ABC) ?

Remember the theorem: the angle formed by the tangent and the chord is half of the measure of the intercepted arc.

Therefore, the arc is double the angle. $$ m\overparen{ABC} = 2 \cdot 110^{\circ}=55^{\circ} $$

Challenge Problems

Problem 3

For the $$ m\overparen{ABC} $$ ? to equal ¾ the total measure of the circle's circumference, what must be the value of X?

Total measure of circle's circumference = 360°.

$$ \frac 3 4 (360^{\circ}) = 270^{\circ} $$

By our theorem, we know that the angle formed by a tangent and a chord must equal half of the intercepted arc so $$x = \frac 1 2 \cdot 270^{\circ} =135^{\circ} $$ .