Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
A curve in geometry is a figure traced out on a plane surface with the help of a sharp pencil without lifting it from the paper. Even a straight line is considered a curve in mathematics. Visualise a doodle or a random scribble on a page; these are all curves.
A Simple Curve is a curve that does not cross itself at any point during its path. For example, a circle or the letter 'S' are simple curves. However, a figure-eight shape is not a simple curve because it intersects itself at the center.
An Open Curve is a curve where the starting point and the ending point are different. Visualise a piece of string with two distinct ends, such as the shape of the letter 'C' or a 'V'.
A Closed Curve is a curve that has no end points and encloses an area. It starts and ends at exactly the same point. Examples include a circle, a triangle, or a closed loop where the path returns to the beginning.
The Interior of a closed curve refers to the area located inside the boundary of the curve. If you imagine a fenced garden, the grass inside the fence is in the interior. In a diagram, a point placed inside a circle is said to be in its interior.
The Exterior of a closed curve refers to the area located outside the boundary of the curve. Any point that is not enclosed by the curve is considered to be in the exterior.
The Boundary of a curve is the actual line or path that defines the shape. A point that lies exactly on the line forming the curve is said to be on the boundary of the curve.
A Region is the combination of the interior of a closed curve and its boundary together. Mathematically, .
📐Formulae
💡Examples
Problem 1:
Classify the following curves as (i) Open or (ii) Closed: (a) A shape like the letter 'W', (b) A shape like the letter 'O', (c) A triangle, (d) A line segment.
Solution:
Step 1: Identify if the start and end points meet. Step 2: For (a), the ends of 'W' are separate, so it is an Open Curve. Step 3: For (b), the path of 'O' returns to the start, so it is a Closed Curve. Step 4: For (c), a triangle is a path of three segments that close back on themselves, so it is a Closed Curve. Step 5: For (d), a line segment has two distinct endpoints, so it is an Open Curve.
Explanation:
Open curves have distinct endpoints, while closed curves form a continuous loop where the start and end points coincide.
Problem 2:
A point is inside a square, a point is on one of the sides of the square, and a point is outside the square. Define the positions of , , and in terms of the curve's properties.
Solution:
Step 1: Determine the location relative to the boundary of the closed curve (the square). Step 2: Since is inside the boundary, is in the Interior of the curve. Step 3: Since lies on the line segment forming the square, is on the Boundary of the curve. Step 4: Since is outside the enclosed area, is in the Exterior of the curve.
Explanation:
Closed curves divide the plane into three parts: the interior (inside), the exterior (outside), and the boundary (the curve itself).