A tangent line segment \overline{PT} touches a circle at point T. A secant line from exter… — ACT

A tangent line segment $\overline{PT}$ touches a circle at point $T$ . A secant line from external point $P$ intersects the circle at points $A$ and $B$ , such that point $A$ lies on segment $\overline{PB}$ . If $m\angle P = 35^\circ$ and the measure of minor arc $AT$ is $50^\circ$ , what is the degree measure of inscribed angle $\angle TAB$ ?

Answer: 60 degrees

Answer

The degree measure of inscribed angle

\angle TAB

is 60 degrees.

According to the exterior angle theorem for circles, the angle formed by a tangent and a secant meeting at an external point

P

is equal to half the difference of the intercepted arcs:

m\angle P = \frac{1}{2}(m\widehat{BT} - m\widehat{AT})

. Substituting

m\angle P = 35^\circ

and

m\widehat{AT} = 50^\circ

into the equation gives

35^\circ = \frac{1}{2}(m\widehat{BT} - 50^\circ)

, which simplifies to

m\widehat{BT} = 120^\circ

. The inscribed angle

\angle TAB

intercepts arc

BT

. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc, giving

m\angle TAB = \frac{1}{2}(120^\circ) = 60^\circ

. Alternatively, inside triangle

PAT

, the tangent-chord angle

PTA

intercepts arc

AT

, so

m\angle PTA = \frac{1}{2}(50^\circ) = 25^\circ

. Since the angles in triangle

PAT

sum to

180^\circ

m\angle PAT = 180^\circ - (35^\circ + 25^\circ) = 120^\circ

. Angle

TAB

is supplementary to angle

PAT

, so

m\angle TAB = 180^\circ - 120^\circ = 60^\circ

Step-by-Step Solution

Use the exterior angle relationship for the secant and tangent to find the measure of arc

BT

m\widehat{BT} = 120^\circ

The exterior angle measure equals half the difference of intercepted arcs

BT

and

AT

35^\circ = \frac{1}{2}(m\widehat{BT} - 50^\circ)

Use the Inscribed Angle Theorem to find

m\angle TAB

m\angle TAB = 60^\circ

An inscribed angle measure is equal to half the measure of its intercepted arc:

m\angle TAB = \frac{1}{2}(120^\circ) = 60^\circ

Key Concept

Secant-Tangent Angle Theorem and Inscribed Angle Theorem

Estimated Time:1m 30s

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