Question

the figure shows a circle inscribed in a triangle. to construct the inscribed circle, angle bisectors were first constructed at each angle of the triangle. which happened next?
a circle was constructed using the intersection of the angle bisectors as the center of the circle and the obtuse vertex as a point on the circumference of the circle.
a circle was constructed using a vertex as the center of the circle and the intersection of the angle bisectors as a point on the circumference of the circle.
segments perpendicular to the sides of the triangle through the intersection of the angle bisectors were constructed.
segments bisecting each side of the triangle were constructed through the intersection of the angle bisectors.

Explanation:

To construct an inscribed circle (incircle) in a triangle:

1. First, construct angle bisectors of the triangle's angles. Their intersection is the incenter, which is equidistant from all sides.
2. Next, we need the radius of the incircle, which is the distance from the incenter to any side of the triangle. To find this, we construct perpendicular segments from the incenter (intersection of angle bisectors) to the sides of the triangle. These perpendiculars give the radius length.

Now, analyze the options:

• Option 1: Using an obtuse vertex as a point on the circumference is incorrect (incircle is tangent to sides, not passing through vertices).
• Option 2: Using a vertex as the center is incorrect (incenter, not a vertex, is the center).
• Option 3: Constructing perpendicular segments from the incenter (intersection of angle bisectors) to the sides matches the next step (to find the radius).
• Option 4: Bisecting sides (midsegments) is for circumcircle or centroid/median, not incircle construction.

Answer:

Segments perpendicular to the sides of the triangle through the intersection of the angle bisectors were constructed.