Angle Formed by a Tangent and a Chord

Angle Formed by a Tangent and a Chord

How to solve problems about the angle formed by a tangent and a chord: formula, proof, examples, and their solutions.

Formula

m(angle formed by a tangent and a chord) = (1/2)*m[intercepted arc]

m∠(red) = (1/2)⋅m[blue arc]

∠(red): angle formed by a tangent and a chord
blue arc: intercepted arc

Proof

Angle Formed by a Tangent and a Chord: Proof of the Formula

Draw a diameter
that starts from the tangent point.

Then the upper red angle
is the inscribed angle of the blue arc.

So m∠(red) = (1/2)⋅m[blue arc].

See the inscribed triangle.

The side of the inscribed triangle
passes through the center of the circle.

So the interior purple angle is a right angle.
And the triangle is a right triangle.

Inscribed right triangle

The right triangle is a 'triangle'.
So m∠(purple) + m∠(green) + m∠(red) = 180.

Interior angles of a triangle

The tangent and the diameter are perpendicular.

So the left side angle of the tangent point
is the purple angle (the right angle).

Tangent to a circle

Since m∠(purple) + m∠(green) + m∠(red) = 180,
the right side angle of the tangent point
is the red angle.

So m∠(right side red angle) = (1/2)⋅m[blue arc]

Example 1

If the measure of arc AB is 110, find the value of x. The measure of the angle formed by a tangent and a chord: x.

m∠(red) = x, m[blue arc] = 110

x = (1/2)⋅110

Example 2

Find the value of x. The measure of the angle formed by a tangent and a chord: x. The measure of the inscribed angle: 80.

m∠(red) = x, m[blue arc]

x = (1/2)⋅m[blue arc]

80 = (1/2)⋅m[blue arc]

Inscribed angle

Both red angles have the same intercepted arc.
(blue arc)

So x = 80.