Question

based on the given diagram, complete the sentence below. answer point d is the of △abc because (overline{de}), (overline{df}), and (overline{dg}) are all

Explanation:

Step1: Recall the definition of the in - center of a triangle

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. Also, the in - center is equidistant from all the sides of the triangle. The perpendicular distance from a point to a side of a triangle is considered. If a point is equidistant from the three sides of a triangle (i.e., the lengths of the perpendiculars from the point to the sides are equal), then that point is the in - center of the triangle.

Step2: Analyze the given segments

In the given diagram, \(\overline{DE}\), \(\overline{DF}\), and \(\overline{DG}\) are perpendicular to the sides of \(\triangle ABC\). Since \(DE = DF=DG\) (implied by the properties of the in - center construction where the distances from the in - center to the sides are equal), and the in - center is the point inside the triangle that is equidistant from all three sides.

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 (by the definition of the in - center).