Question

which statement about (overleftrightarrow{bc}) is correct? (overleftrightarrow{bc}) is not a tangent line because (mangle abc
eq 90^circ); (overleftrightarrow{bc}) is a tangent line because (\triangle abc) is a right triangle; (overleftrightarrow{bc}) is a tangent line because (mangle abc = 90^circ); (overleftrightarrow{bc}) is a tangent line because (mangle abc) is acute.

Explanation:

To determine if \(\overleftrightarrow{BC}\) is a tangent, we use the tangent - radius theorem: A line is tangent to a circle if and only if it is perpendicular to the radius at the point of tangency (i.e., the angle between the radius and the line is \(90^{\circ}\)).

1. Analyze the triangle \(ABC\): We know that in \(\triangle ABC\), the sum of interior angles is \(180^{\circ}\). Given \(\angle BAC = 25^{\circ}\) and \(\angle ACB=65^{\circ}\), then \(\angle ABC=180^{\circ}-\angle BAC - \angle ACB=180 - 25-65 = 90^{\circ}\).
2. Apply the tangent - radius theorem: Since \(AB\) is a radius (as \(A\) is the center and \(B\) is on the circle) and \(\angle ABC = 90^{\circ}\) (so \(BC\perp AB\) at the point of tangency \(B\)), \(\overleftrightarrow{BC}\) is a tangent line because \(m\angle ABC = 90^{\circ}\).

Let's analyze the other options:

• Option 1: \(m\angle ABC = 90^{\circ}\), so this statement is wrong.
• Option 2: Just because \(\triangle ABC\) is a right triangle does not mean \(BC\) is tangent. The right angle must be at the point of tangency between the radius and the line. But in this case, the right angle is at \(B\) (the point of tangency) and \(AB\) is the radius, so the correct reason is \(m\angle ABC = 90^{\circ}\) (perpendicular to radius), not just that \(\triangle ABC\) is right - angled.
• Option 4: The fact that \(\angle ABC\) is acute is wrong because we calculated \(\angle ABC = 90^{\circ}\), and also the acuteness of the angle has nothing to do with the tangent property.

Answer:

\(\overleftrightarrow{BC}\) is a tangent line because \(m\angle ABC = 90^{\circ}\) (the third option)