How Do I Find the Area of a Triangle with Non-Intersecting Vertices?

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Discussion Overview

The discussion revolves around finding the area of a triangle whose vertices do not align with the intersections of horizontal and vertical lines. Participants explore various methods for calculating the area, particularly in the context of a math competition for elementary students.

Discussion Character

  • Exploratory, Homework-related, Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in determining the side lengths of the triangle due to the non-intersecting vertices.
  • Another participant suggests drawing a tight rectangle around the triangle and estimating its side lengths, proposing that the area can be calculated by subtracting the areas of three right triangles from the rectangle's area.
  • A subsequent reply echoes the rectangle approach but notes that the triangle's vertices do not touch the rectangle, indicating potential complications.
  • Further, a participant proposes drawing a rectangle that touches the triangle's vertices, suggesting it should be parallel to the grid lines, and introduces the idea of using a help line to facilitate the area calculation by aligning with right triangles.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of using a rectangle for area estimation, with some acknowledging complications due to the triangle's vertices not touching the rectangle. The discussion remains unresolved regarding the best approach to calculate the area.

Contextual Notes

Participants highlight limitations related to the alignment of the triangle's vertices with the grid, which affects the proposed methods for area calculation.

Monoxdifly
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What's the area of the triangle? It's hard because the vertices aren't in the intersections of horizontal and vertical lines, so I have a hard time determining the side lengths, and it's also for Elementary Students Math Olympiads too.
 

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Hi Mr. Fly,

How about we draw a tight rectangle around the triangle and we estimate its side lenghts?
The area of the triangle is then the area of the rectangle minus the area of the three right triangles.
 
Klaas van Aarsen said:
Hi Mr. Fly,

How about a tight rectangle around the triangle and estimate its side lenghts?
The area of the triangle is then the area of the rectangle minus the area of the three right triangles.

But the vertices of the triangle don't even touch the rectangle, so I think it's not that easy.
 
Monoxdifly said:
But the vertices of the triangle don't even touch the rectangle, so I think it's not that easy.

Draw a rectangle that touches the vertices. It won't be on the grid lines - only parallel to them.

Alternatively we can draw a help line from A downward to a point D on the same grid line as C.
Now we can add and subtract the relevant right triangles that are aligned with the help lines.
 

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