Properties of Circle GMAT Problem Solving

Topic: Properties of Circle

1. Definition
2. Important terms
3. Important properties - Angles and Lines
4. Important Circle formula

‘Properties of Circle’ is the topic from the GMAT Quantitative problem set. GMAT quantitative reasoning section tests the candidate's ability to solve mathematical and quantitative problems, interpret graph data, and mathematical reasoning. The quantitative section of the GMAT exam consists of 31 questions. This topic from the GMAT quant section has description of the properties of circle. The topic describes and illustrates images showing the properties of the circle. In this Problem-solving question type, candidates' logical and analytical reasoning skills are checked.

Purpose of the article:

In this article,

• We will get an idea about circles and the related terms.
• We will also get to know about the properties of circles.

Circle- Definition

When all points that are at a fixed distance from a fixed point are joined, the geometrical figure obtained is called a circle. We can also define a circle as a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre).

Important Terms Related to Circle

 

1. Center - The fixed point in the circle is called the center. So, the distance between the center and any point on the circumference.
2. Radius - Radius is the fixed distance between the center and the set of points. It is denoted by “R.” It is also defined as the distance between the center and any point on the circumference is called the radius of the circle.
3. Diameter - Diameter is a line segment, having its endpoints on the circle and passing through the center.

So, basically, a diameter can be broken into two parts:

1. One part from one endpoint of the diameter to the center of the circle.
2. And, the other part from the center of the circle to another endpoint of the diameter.
    

Hence, the formula for the Diameter = Twice the length of the radius or “D= 2R.”

4. Circumference - Circumference is defined as the measurement of the outside boundary of the circle.

• So, the length of the circle or the perimeter of the circle is called circumference.

 

5. Arc - An arc is a portion of the circle’s circumference.

From any two-point that lie on the boundary of the circle, two arcs can be created: A Minor and a Major Arc.

• Minor arc: The shortest arc created by two points.
• Major Arc: The longest arc created by two points.

6. Sector - It is formed by joining the endpoints of an arc with the center of the circle.

If we join the endpoints with the center, two sectors will be obtained:

• Minor
• Major.
   

By default, we only consider the Minor sector unless it is mentioned otherwise.

 

7. Semi-circle

• A semi-circle is half part of the circle or,
• A semi-circle is obtained when a circle is divided into two equal parts or
• A semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180°

 

Important properties of Lines in a Circle:

1. Chord - A chord is a line segment whose endpoints lie on the boundary of the circle. The diameter is the largest chord in a circle.

Properties of Chord:

• Perpendicular dropped from the center divides a chord into two equal parts.
• Chords of a circle, equidistant from the center of the circle are equal

 

2. Tangent - A tangent is a line that touches the circle at any one point.

Properties of Tangent:

• Radius is always perpendicular to the tangent at the point where it touches the circle.
• The tangent touches the circle at a single point.
• The tangent will never intersect the circle at two points.
• The length of tangents from an external point to a circle are equal.

 

Important Properties of angles in a circle

1. Inscribed Angle - An inscribed angle is the angle formed between two chords when they meet on the boundary of the circle.

Properties of Inscribed Angle

• All the angles formed by an arc on the circumference of the circle are always equal.
• Angle in a semi-circle is always 900.

Important Formulas: Area and Perimeter

Perimeter:

• Perimeter or Circumference of a circle = 2 × π × R.
• Length of an Arc = \(\frac{Central angle made by the arc}{360^0} × 2 × π × R\)

Area:

• Area of a circle = π × R^2
• Area of a sector =\( \frac{Central angle made by the arc}{360^0} × π × R^2\)

Key takeaways:

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