Question

jace is making a water play table from a triangular table top. he sketched a dotted line to show where he wants to add a circular water bowl. it should go all the way to the edges of the table as shown.
image of triangle with vertices a, b, c; side ab = 3.5 feet, side bc = 4 feet, side ac = 2 feet; angle at b is 30°, angle at c is 60°; a dotted circle is inside the triangle
what should jace do first to find the exact location of the center of the circular water bowl?
○ he should find the midpoint of each edge of the table.
○ he should bisect each of the angles at the vertices of the triangular table top.
○ he should draw perpendicular lines through any point along the edges of the table.
○ he should draw perpendicular line segments from each corner to the opposite edge of the table.

Explanation:

To find the center of a circle inscribed in a triangle (incircle), we need the incenter. The incenter is the intersection of the angle bisectors of the triangle's vertices. Let's analyze each option:

• Option 1: Finding midpoints of edges gives midsegments or perpendicular bisectors (for circumcenter), not incenter.
• Option 2: Bisecting each angle of the triangle's vertices. The incenter (center of the incircle) is where angle bisectors meet. This is correct for finding the center of a circle tangent to all three sides (the water bowl touching all edges).
• Option 3: Drawing perpendicular lines through any point on edges is not a method to find the incenter.
• Option 4: Drawing perpendicular segments from corners to opposite edges (altitudes) finds the orthocenter, not the incenter.

So the correct method is bisecting each angle at the vertices.

Answer:

He should bisect each of the angles at the vertices of the triangular table top.