Question

three friends want to meet at a place that is the same distance from each of their houses. they draw a map and measure the approximate distances and angles, as shown.

which step should they take next to find a meeting place that is the same distance from each persons house?

find the midpoint of the segments which connect their houses.
bisect each of the angles with vertices at their houses.
draw perpendicular lines through any point on the segments which connect their houses.
draw perpendicular line segments from each vertex to the opposite side.

Explanation:

Identify the geometric requirement

The meeting place must be equidistant from all three houses, which form the vertices of a triangle. The point equidistant from the vertices of a triangle is the circumcenter.

Determine the construction method for the circumcenter

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. A perpendicular bisector of a segment is a line perpendicular to the segment passing through its midpoint.

Evaluate the given options

• Finding only the midpoints does not locate the intersection point.
• Bisecting the angles finds the incenter (equidistant from the sides, not the vertices).
• Drawing perpendicular lines through any point does not guarantee they pass through the midpoints.
• Drawing perpendicular line segments from each vertex to the opposite side finds the orthocenter.
• The option "Draw perpendicular lines through any point on the segments which connect their houses" is a slightly simplified description of constructing perpendicular bisectors (specifically, drawing perpendicular lines through the midpoints, which are points on the segments).

Answer:

• Find the midpoint of the segments which connect their houses.
• Bisect each of the angles with vertices at their houses.
• Draw perpendicular lines through any point on the segments which connect their houses. (Correct answer)
• Draw perpendicular line segments from each vertex to the opposite side.