Problem

Construct the incircle of triangle \(ABC\). 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 ABC with vertices A, B, and C labeled.

Big Picture

What this problem is really about

An incircle center is chosen by equal distances to the side lines, so angle bisectors are the correct loci. We’ll intersect two of them, drop a perpendicular to any side to define the radius, and use equal side distances with radius-tangent perpendicularity to justify all three contacts.

Answer choices

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Preview only

Construct the bisectors of ∠A and ∠B. Their intersection I is inside the triangle; use its perpendicular distance to a side as the radius and draw the circle centered at I.

Construct the perpendicular bisector of AB and the bisector of ∠B, then use their intersection as the incenter.

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

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

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Construct the incircle of triangle \(ABC\). 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.

What this problem is really about

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