The city planning commission is looking to put in sidewalks where the three medians of the triangle are located. What would be true about the sidewalks if the model of the city par... The city planning commission is looking to put in sidewalks where the three medians of the triangle are located. What would be true about the sidewalks if the model of the city park is an equilateral triangle? Select all that apply.
Understand the Problem
The question is asking about the properties of the medians of an equilateral triangle, specifically what can be concluded regarding sidewalks placed along these medians in the context of a triangular city park.
Answer
The sidewalks would bisect the angles of the park and would meet in a single location.
Answer for screen readers
- The sidewalks would bisect the angles of the park.
- The sidewalks would meet in a single location.
Steps to Solve
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Understanding Medians of an Equilateral Triangle
In an equilateral triangle, the three medians are the segments connecting each vertex to the midpoint of the opposite side. They bisect the angles at each vertex. -
Analyzing the Properties of Medians
Since the triangle is equilateral, all three medians are equal in length. Each median also bisects the triangle into two smaller triangles of equal area. -
Examining Angle Bisectors
Because the medians also act as angle bisectors in an equilateral triangle, they will bisect the angles at the vertices of the park. -
Considering Meeting Point of Medians
All three medians of an equilateral triangle meet at a single point known as the centroid, which divides each median into two segments with a 2:1 ratio. -
Analyzing the Total Length of the Sidewalks
The total length of the sidewalks (the combined length of the medians) does not equal the perimeter of the triangle, which is the distance around the park. -
Dividing the Triangle
The sidewalks (medians) divide the triangle into 6 smaller triangles, but they are not noncongruent. Each of these smaller triangles has the same area but different shapes based on the positions of the vertices.
- The sidewalks would bisect the angles of the park.
- The sidewalks would meet in a single location.
More Information
In an equilateral triangle, the medians have unique properties: they bisect the vertex angles, are equal in length, and converge at the centroid. They do not create noncongruent triangles, and the total length of the medians is not equal to the perimeter.
Tips
- Confusing medians with angle bisectors in non-equilateral triangles, which do not have the same properties.
- Misunderstanding that the medians can create noncongruent triangles; they always produce congruent areas within an equilateral triangle.
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