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 problem asks for a point that is equidistant (the same distance) from three given points (the houses of Jo, Dom, and Carrie), which form the vertices of a triangle.

Apply the circumcenter theorem

The point equidistant from all three vertices of a triangle is the circumcenter. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle.

Evaluate the given options

• Finding the midpoints only does not locate the intersection point.
• Bisecting the angles finds the incenter, which is equidistant from the sides, not the vertices.
• Drawing perpendicular lines through the midpoints (which are points on the segments connecting their houses) represents constructing the perpendicular bisectors.
• Drawing perpendicular line segments from each vertex to the opposite side finds the orthocenter.

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.