Decide whether the candidate line is tangent using this complete coordinate evidence: \(T\) is stated to lie on \(\text{both}~\text{the}~\text{circle}~\text{and}~\text{candidate}~\text{line}\) \(y=2x+1\); radius \(OT\) has slope \(-1/2\).

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No. The radius slope is 1/2, so its product with 2 is 1.

No. Adding the slopes gives 3/2, so the lines are not perpendicular.

Yes. T is on both figures, and (−1/2)(2) = −1, so the line is perpendicular to OT and tangent at T.

No. The reciprocal of −1/2 is −2, so line slope 2 is not perpendicular.

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Curriculum context

Course: Math II
Standard: G-C.4
Category: Geometry
Domain: Circles
Objective: Construct a tangent line from an external point to a circle.
Problem type: Verify tangency from coordinate incidence and perpendicularity