Question

5 problem campout
oscar purchased land on which to build a campground. use your geometry skills to help oscar make
decisions on how best to use his camps. determine which point of concurrency should be used for each
response. justify your response. (2 pts each)
points of concurrency
centroid circumcenter incenter orthocenter
(figures of triangles with different points of concurrency)
5 the building plan contains a central area with a pool, playground and basketball courts as shown.
oscar plans to build a bathroom that is equidistant from the basketball courts, the playground and the
pool. which point of concurrency of a triangle should oscar use to determine the location of the
bathroom? explain your reasoning.
(handwritten: circumcenter)
(figure: triangle with vertices labeled playground, basketball courts, pool)
6 on another part of the campground, oscar plans to build a fire pit between three roads that
intersect. he is going to make trails from the fire pit to each road, and wants each of the trails to
the same distance. which point of concurrency of a triangle should oscar use to determine the
location of the fire pit? explain your reasoning.
(handwritten: incenter because it... (text cut off))
(figure: three intersecting lines labeled snore drive, lake road, oak lane)
© oscar is going to build a community grill. the grill is triangular, build on a single base. which point
of concurrency of a triangle should oscar use for the location of the base so that the grill is most
balanced and stable? explain your reasoning.
(handwritten: centroid)
(figure: a triangular shape with parallel lines)

Explanation:

Part (a)

To determine the location of the bathroom equidistant from three points (basketball courts, playground, pool), we use the property of the circumcenter. The circumcenter of a triangle is the point equidistant from all three vertices (since it's the center of the circumscribed circle passing through all vertices). So, the circumcenter is the correct point of concurrency here.

For a fire pit equidistant from three roads (which form the sides of a triangle), we use the incenter. The incenter is the point where the angle bisectors of a triangle meet, and it is equidistant from all three sides of the triangle (as it's the center of the inscribed circle tangent to all sides). Thus, the incenter is the right choice.

To make the triangular grill balanced and stable, we use the centroid. The centroid is the point of concurrency of the medians of a triangle, and it represents the center of mass (or balance point) of the triangle. Placing the base at the centroid - related position (or using the centroid's properties) ensures the grill is balanced as the centroid is where the triangle would balance perfectly.

Answer:

Circumcenter

Part (b)