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Geometry - Basic Geometrical Ideas: Points, Lines, Angles, Curves, Polygons

Grade 6ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A point indicates a position and has no dimensions, while a line extends infinitely in both directions. A line segment is a fixed portion of a line with two endpoints, and a ray starts at one point and goes on forever in one direction.

Comparison of line, line segment, and ray.
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An angle is formed by two rays sharing a common endpoint called the vertex. The rays are the arms of the angle.

An angle showing vertex and arms.
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Curves can be open or closed. A simple closed curve that does not cross itself is the basis for defining polygons. A polygon is a closed figure made entirely of line segments.

Comparison between an open curve and a closed polygon.
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A circle is a simple closed curve where every point on the boundary is at an equal distance (radius) from a fixed center point.

Circle with center O and radius marked.

📐Formulae

d=2×rd = 2 \times r (Diameter is twice the radius)

r=d2r = \frac{d}{2} (Radius is half the diameter)

Sum of angles in a triangle=180∘\text{Sum of angles in a triangle} = 180^{\circ}

Sum of angles in a quadrilateral=360∘\text{Sum of angles in a quadrilateral} = 360^{\circ}

💡Examples

Problem 1:

If the radius of a circular wheel is 14 cm14 \text{ cm}, find the length of its diameter.

Solution:

  1. Identify the given value: Radius r=14 cmr = 14 \text{ cm}.
  2. Use the relationship formula: d=2×rd = 2 \times r.
  3. Substitute the value: d=2×14 cmd = 2 \times 14 \text{ cm}.
  4. Calculate the result: d=28 cmd = 28 \text{ cm}.

Explanation:

Since the diameter of any circle is exactly twice the length of its radius, we multiply the given radius by 22 to find the diameter.

Problem 2:

In a quadrilateral PQRSPQRS, how many diagonals can be drawn? Name them.

Solution:

  1. A quadrilateral is a polygon with 44 vertices: P,Q,R,P, Q, R, and SS.
  2. Diagonals connect non-adjacent vertices.
  3. Identify pairs of non-adjacent vertices: (P,R)(P, R) and (Q,S)(Q, S).
  4. Therefore, there are 22 diagonals: PR‾\overline{PR} and QS‾\overline{QS}.

Explanation:

In any polygon, a diagonal is a line segment connecting two corners that are not next to each other. For a four-sided figure, only two such pairs exist.

Problem 3:

Identify the number of sides and vertices in the given polygon ABCDEABCDE. Also, name any two adjacent sides.

A pentagon labeled ABCDE.

Solution:

  1. The polygon has 55 sides: ABAB, BCBC, CDCD, DEDE, and EAEA.
  2. It has 55 vertices: AA, BB, CC, DD, and EE.
  3. Two adjacent sides are ABAB and BCBC (they meet at vertex BB).

Explanation:

A polygon with five sides is called a pentagon. Adjacent sides are any two sides that share a common endpoint (vertex).

Problem 4:

In the following figure, identify the rays and the angle formed at the vertex OO.

Angle AOB formed by rays OA and OB.

Solution:

  1. The two rays are OA⃗\vec{OA} and OB⃗\vec{OB}.
  2. The vertex is OO.
  3. The angle formed is ∠AOB\angle AOB or ∠BOA\angle BOA.

Explanation:

Rays start from a fixed point (origin) and extend infinitely. When two rays originate from the same point, the space between them is measured as an angle.