Dart Game

Kritima Lamichhane - Caviness 5th

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Background

Sally, John, and Jake were playing with darts in their back yard. They were standing in a triangle shape 8 feet apart from each other and have a cylinder shaped pole, with a balloon on top, which is their target. They need to place the balloon target in the center of the triangle so that it is equal distance from each friend. Where should they put their target so that it is equidistant from the three friends?

Picture Depiction of where Sally, John, & Jake are Standing

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Point of Concurrency shown with all Labels

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What each color represents:

Blue - the equilateral triangle

Pink - arcs created by the 3 sides of the triangle

Green - lines of intersection that show the circus center

Orange - Circle that proves the points are equidisant

Purple - lines connecting vertices to the circumcenter

Midpoint, Slope, & Equation of the Lines

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Steps to solve the problem:

I started off my drawing the graph, and labeling the 3 points, that each represented one of the kids. I connected the lines and created an equilateral triangle (4 inches on each side - each inch represented 2 feet). Then I set my compass a little wider than half of a line, and placing the tip of the compass on point A, I drew an arc. Keeping the compass in the same shape, I drew arcs from point B. Then I drew a line between the intersection of the two arcs. Then I drew an arc from point C. I drew a line through the intersection made by the arcs from point B & C. Where the two lines intersected was the circumcenter.
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Finally

As shown I opened the compass from the circumcenter to one of the points, and then I drew a circle starting the point, the circle hit the other vertices as well. That proves that the circumcenter is equidisant from each vertex. That spot (0,1) is where the kids should place their pole with the balloon that they are all trying to aim for. They will each be throwing their darts the same distance, if they place their target in the circumcenter. That way it is fair when the kids aim for the balloon.
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