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

In summary, the conversation revolves around finding the area of a triangle. One suggestion is to draw a tight rectangle around the triangle and estimate its side lengths. The area of the triangle can then be calculated by subtracting the area of the three right triangles from the area of the rectangle. However, it is noted that the vertices of the triangle may not touch the rectangle, making it more challenging. Another suggestion is to draw a help line from one of the vertices downwards to a point on the same grid line as another vertex, and then add or subtract relevant right triangles to find the area.
  • #1
Monoxdifly
MHB
284
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

1. How do I find the area of a triangle with non-intersecting vertices using the Pythagorean theorem?

To find the area of a triangle with non-intersecting vertices using the Pythagorean theorem, you will need to know the lengths of all three sides of the triangle. Once you have the lengths, you can use the formula A = 1/2 * base * height, where the base is one of the sides and the height is the perpendicular distance from that side to the opposite vertex. You can then use the Pythagorean theorem (a² + b² = c²) to find the height, and plug it into the formula to calculate the area.

2. Can I use the Heron's formula to find the area of a triangle with non-intersecting vertices?

Yes, you can use the Heron's formula to find the area of a triangle with non-intersecting vertices. This formula is especially useful when you do not know the lengths of all three sides of the triangle. The formula is A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (half of the perimeter) and a, b, and c are the lengths of the sides. This formula works for all types of triangles, including those with non-intersecting vertices.

3. What is the difference between the base and height of a triangle with non-intersecting vertices?

The base of a triangle with non-intersecting vertices is one of the sides of the triangle, while the height is the perpendicular distance from that side to the opposite vertex. The base and height are always perpendicular to each other, and they form a right triangle with one of the sides of the triangle. The base and height are essential in calculating the area of a triangle using the formula A = 1/2 * base * height.

4. How do I find the area of a triangle with non-intersecting vertices if I only know the coordinates of the vertices?

If you only know the coordinates of the vertices, you can use the formula A = 1/2 * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| to find the area of the triangle. In this formula, (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three vertices. This formula is derived from the cross-product of two vectors and works for all types of triangles, including those with non-intersecting vertices.

5. Can I use trigonometry to find the area of a triangle with non-intersecting vertices?

Yes, you can use trigonometry to find the area of a triangle with non-intersecting vertices. You will need to know the lengths of at least two sides and the angle between them. You can then use the formula A = 1/2 * a * b * sin(C), where a and b are the lengths of the two sides and C is the angle between them. This formula is derived from the formula for the area of a parallelogram and works for all types of triangles, including those with non-intersecting vertices.

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