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

# Chord, Tangent and the Circle

The Intersection of a Tangent and Chord The Theorem: An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc This means that the measure of arc ABC (the purple portion of the circle itself) is twice the measure of angle C.

Note: Like inscribed angles, when the vertex is on the circle itself, the angle formed is half the measure of the intercepted arc.

### Practice Problems

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 170°.

Therefore $$x = \frac{1}{2} \cdot 170 = 85^{\circ}$$

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. $$m\overparen{ABC} = 2 \cdot 70^{\circ}=140^{\circ}$$

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 $$x = \frac 1 2 \cdot 270^{\circ} =135^{\circ}$$

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

The key to this problem is recognizing that $$\overline{AOC}$$ is a diameter .

Therefore, $$m \overparen{ABC} = 180 ^{\circ}$$

At this point, you can use the formula, $$\\ m \angle ACZ= \frac{1}{2} \cdot 180 ^{\circ} \\ m \angle ACZ= 90 ^{\circ}$$

The key to this problem is recognizing that the total degrees in a circle is $$360^{\circ}$$

From there you can set up an equation using the 3: 2 ratio.

$$3x + 2x =360 \\ 5x = 360 \\ \frac{5x}{5} = \frac{360}{5} \\ x = 72 \\ \overparen {JLM }= 2x = 2 \cdot 72 \\ \overparen {JLM }=144 ^{\circ}$$

At this point, you can use the formula, $$\\ m \angle MJK= \frac{1}{2} \cdot 144 ^{\circ} \\ m \angle MJK = 72 ^{\circ}$$

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