Question

based on the given diagram, complete the sentence below.
answer attempt 1 out of 2
point ( d ) is the of ( \triangle a b c ) because ( overline{d e}, overline{d f} ), and ( overline{d g} ) are all

Explanation:

Step1: Recall definitions

A point that is equidistant from the sides of a triangle is the in - center. The in - center is the intersection of the angle bisectors of a triangle.

Step2: Analyze the given segments

Since \(DE\), \(DF\), and \(DG\) are perpendicular to the sides of \(\triangle ABC\) (they are shown as right angles in the diagram) and if a point is at a perpendicular distance (the length of these segments) from the sides of a triangle, and these distances (the lengths of \(DE\), \(DF\), \(DG\)) are equal (by the property of the in - center), then the point \(D\) is the in - center.

Answer:

Point \(D\) is the in - center of \(\triangle ABC\) because \(\overline{DE}\), \(\overline{DF}\), and \(\overline{DG}\) are all perpendicular distances from \(D\) to the sides of the triangle (and are equal as per the in - center property).