Problem

In triangle \(ABC\), incenter \(I\) has perpendicular foot \(T\) on \(AB\) and \(IT~=~4~\text{cm}\). State the incircle's center and radius, identify its tangency point on \(AB\), and use the relationship between \(IT\) and \(AB\) to justify that tangency.

Triangle ABC with incenter I, perpendicular foot T on side AB, and segment IT labeled 4 cm; no incircle is shown.

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What this problem is really about

For an incenter, the radius is not a segment to a vertex; it is a perpendicular distance to a side. We’ll read the named perpendicular foot, use that segment as the circle’s radius, and apply the radius-tangent converse at its endpoint.

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The incircle is centered at I with radius 4 cm because IT is parallel to AB.

The incircle is centered at T with radius 4 cm.

The incircle is centered at I with radius 2 cm because IT is the diameter.

The incircle is centered at I with radius IT = 4 cm, and it is tangent to AB at T because IT ⟂ AB.

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In triangle \(ABC\), incenter \(I\) has perpendicular foot \(T\) on \(AB\) and \(IT~=~4~\text{cm}\). State the incircle's center and radius, identify its tangency point on \(AB\), and use the relationship between \(IT\) and \(AB\) to justify that tangency.

What this problem is really about

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