Problem

Construct the incircle of triangle \(DEF\). Name the angle bisectors used to locate \(I\), explain why \(I\) has equal perpendicular distances to \(\text{all}~\text{three}~\text{sides}\), construct \(\text{one}~\text{perpendicular}~\text{foot}\) to set the radius, and state the resulting tangencies.

Unworked triangle DEF with vertices D, E, and F labeled.

Answer choices

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Construct the bisectors of ∠D and ∠E. Their intersection I has equal perpendicular distances to DE, EF, and DF; use one perpendicular distance as the radius. The circle centered at I is tangent to all three sides.

Construct only the bisector of ∠D and choose a point on it as I.

Construct the perpendicular bisector of DE and the bisector of ∠E, then use their intersection as the incenter.

Construct the bisectors of ∠D and ∠E, then use the distance from I to a vertex as the radius.

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Construct the incircle of triangle \(DEF\). Name the angle bisectors used to locate \(I\), explain why \(I\) has equal perpendicular distances to \(\text{all}~\text{three}~\text{sides}\), construct \(\text{one}~\text{perpendicular}~\text{foot}\) to set the radius, and state the resulting tangencies.

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