Question

in circle ( p ), ( overline{pa} ) is a radius and ( overline{ab} ) is a tangent segment. which statement must be true? ( angle pab ) is an acute angle. ( pa = ba ) ( pa = pb ) ( angle pab ) is a right angle.

Explanation:

Step1: Recall the tangent - radius theorem

The tangent - radius theorem states that a tangent to a circle is perpendicular to the radius at the point of tangency.
In circle \(P\), \(\overline{AB}\) is a tangent segment and \(\overline{PA}\) is a radius. The point of tangency is \(A\).
So, \(\overline{PA}\perp\overline{AB}\).

Step2: Analyze the angle \(\angle PAB\)

If two lines are perpendicular, the angle between them is \(90^{\circ}\).
Since \(\overline{PA}\perp\overline{AB}\), by the definition of perpendicular lines, \(\angle PAB = 90^{\circ}\).
An acute angle is an angle with a measure between \(0^{\circ}\) and \(90^{\circ}\), so \(\angle PAB\) is not an acute angle.
For the lengths, \(PA\) is a radius, \(BA\) is a tangent segment. In general, \(PA
eq BA\) (except in some special - constructed non - standard cases which are not the norm in basic circle - tangent relations). Also, \(PB\) is the hypotenuse of the right - triangle \(PAB\) (\(PB=\sqrt{PA^{2}+AB^{2}}\) by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = PA\), \(b = AB\) and \(c = PB\)), so \(PA
eq PB\).

Answer:

\(\angle PAB\) is a right angle.