Question

Find the area triangle formed by the given three points. \((1,1),(3,3),\) and (3,3)

Short answer

The given points do not form a triangle since two points are identical.

Step-by-step solution

Check for Distinct Points

First, observe the given points: \((1, 1)\), \((3, 3)\), and \((3, 3)\). Notice that two of the points \((3, 3)\) are identical. To form a triangle, all three points must be distinct.

Conclude No Triangle Can Be Formed

Since two of the points are the same, this means the points do not form a triangle. A triangle requires three distinct points to create three sides. With two points being identical, only a line or a point is formed, depending on their placement.

Key concepts

Triangle Formation

For a shape to be considered a triangle in geometry, it is essential that there are exactly three distinct points. These points serve as the triangle's vertices, and the straight lines connecting these points form the sides of the triangle. These points must not only be distinct but also not collinear (more on this later). Without three separate points, it's impossible to form the three sides needed, which is the foundation of any triangle.

Triangles are one of the simplest yet fundamental shapes in geometry. They have various properties and types based on angle measures and side lengths, such as equilateral, isosceles, and scalene triangles. However, regardless of their type, all triangles share the necessity of having three corners and three sides, a simple yet essential condition for their formation.

Distinct Points in Geometry

In geometry, the concept of 'distinct points' is crucial for shape formation. Distinct points mean different locations on a plane or in space, each separated from one another. When forming geometric shapes such as triangles, distinct points ensure that a shape is well-defined with clear boundaries. It's important to remember:

• Each point represents a unique position.
• No two points can occupy the same space to remain distinct.
• Having non-distinct points leads to degenerate shapes (e.g., a line or a point rather than a triangle).

In our exercise, the points given were \(1, 1\), \(3, 3\), and \(3, 3\). Since two of these points are identical, they cannot form a triangle as there's no third distinct vertex. Always ensure distinctness when analyzing or plotting points to form geometric shapes.

Collinear Points

Collinear points are points that lie on the same straight line. In triangle geometry, if three points are collinear, they cannot form a triangle. This is because a triangle requires three sides, whereas collinear points would simply lie along a single line.

Here's what you need to understand about collinearity:

• Three points \(A, B, \, \text{and} \, C\) are collinear if the slope between \(A \, \text{and} \, B\) is the same as the slope between \(B \, \text{and} \, C\).
• If points are collinear, they create a degenerate triangle with zero area.
• To form a valid triangle, the points must not all lie on the same line; they must form a closed shape.

When assessing points, it's not just important to ensure they are distinct, but also to verify they are non-collinear to form proper triangles.