Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided figure where all vertices lie on a single circle, known as the circumcircle. One of the most critical properties is that the sum of each pair of opposite angles is always 180 degrees. This unique characteristic makes cyclic quadrilaterals a fundamental concept in geometry, aiding in solving various complex problems.

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    Cyclic Quadrilateral Definition

    A cyclic quadrilateral is a four-sided polygon where all four vertices lie on a single circle. This circle is known as the circumcircle, and its centre is called the circumcentre. Cyclic quadrilaterals have unique properties that make them an interesting topic within geometry.

    Properties of Cyclic Quadrilaterals

    Some key properties of cyclic quadrilaterals include:

    • Opposite Angles: The sum of the measures of opposite angles in a cyclic quadrilateral is always 180 degrees, i.e., if the quadrilateral is ABCD with vertices A, B, C, and D lying on the circumcircle, then \(\theta_A + \theta_C = 180°\) and \(\theta_B + \theta_D = 180°\).
    • Perpendicular Bisectors: The perpendicular bisectors of all sides of a cyclic quadrilateral meet at the circumcentre.
    • Ptolemy's Theorem: For a cyclic quadrilateral with vertices A, B, C, and D, Ptolemy's theorem states: \(AC \times BD = AB \times CD + AD \times BC\).

    Example: Consider a cyclic quadrilateral ABCD inscribed in a circle with vertices A, B, C, and D.

    If AB = 5 cm, BC = 7 cm, CD = 4 cm, and DA = 6 cm, then the lengths of the diagonals can be calculated using Ptolemy's theorem:

    \(AC \times BD = AB \times CD + AD \times BC\)

    By substituting the given values:

    \(AC \times BD = 5 \times 4 + 6 \times 7 = 70\)

    Hint: Remember that the sum of the opposite angles in a cyclic quadrilateral is always 180 degrees!

    A deeper exploration of cyclic quadrilaterals reveals their relationships with other geometric figures. For instance, any rectangle is a cyclic quadrilateral because its opposite angles are right angles, summing up to 180 degrees. Additionally, Brahmagupta's formula can be used to find the area of a cyclic quadrilateral when the lengths of all sides are known:

    \(K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\)

    where \(s\) is the semiperimeter given by \(s = \frac{a+b+c+d}{2}\) and \(a, b, c, d\) are the sides of the quadrilateral.

    Properties of Cyclic Quadrilateral

    Cyclic quadrilaterals have several intriguing properties that distinguish them from other types of quadrilaterals. Understanding these properties deepens your knowledge of geometry.

    Opposite Angles in a Cyclic Quadrilateral

    Opposite Angles Property: In a cyclic quadrilateral, the sum of each pair of opposite angles is always 180 degrees. Mathematically, if the quadrilateral is labelled as ABCD, then \(\theta_A + \theta_C = 180°\) and \(\theta_B + \theta_D = 180°\).

    Example: Suppose you have a cyclic quadrilateral ABCD. If the measure of angle A is 70°, what will be the measure of angle C?

    Since \(\theta_A + \theta_C = 180°\):

    \[\theta_C = 180° - 70° = 110°\]

    Hint: Use the property of opposite angles summing to 180° to quickly check if a quadrilateral is cyclic!

    Deep Dive: The property of opposite angles summing to 180° can be proved using the concept of inscribed angles. Consider the cyclic quadrilateral ABCD inscribed in a circle. Angles A and C subtend the same arc, hence their sum is the same as twice the inscribed angle subtended by the arc. Thus, \(\theta_A + \theta_C = 180°\). This geometric property has fascinating implications in many advanced fields of study, including trigonometry and calculus.

    Cyclic Quadrilateral Theorem

    Ptolemy's Theorem is a fundamental theorem that applies specifically to cyclic quadrilaterals. This theorem provides a relationship between the sides and diagonals of a cyclic quadrilateral.

    Ptolemy's Theorem states:

    For a cyclic quadrilateral with vertices A, B, C, and D, the relation between the sides and diagonals is given by:

    \[AC \times BD = AB \times CD + AD \times BC\]

    Example: Consider a cyclic quadrilateral with sides AB = 8 cm, BC = 6 cm, CD = 5 cm, and DA = 7 cm. Let diagonal AC = 9 cm. Calculate the length of diagonal BD using Ptolemy's theorem.

    Using Ptolemy's theorem:

    \[9 \times BD = 8 \times 5 + 7 \times 6\]

    Simplifying:

    \[9 \times BD = 40 + 42\]

    \[9 \times BD = 82\]

    \[BD = \frac{82}{9} \approx 9.11\] cm

    Deep Dive: Beyond Ptolemy's Theorem, there are other interesting properties and theorems related to cyclic quadrilaterals. For example, the Brahmagupta's formula is used to calculate the area of a cyclic quadrilateral when the lengths of all its sides are known. It is given by:

    \[K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\]

    where \(s\) is the semiperimeter, given by:

    \[s = \frac{a+b+c+d}{2}\]

    and \(a, b, c, d\) are the lengths of the sides. Though you will seldom need this formula early in your studies, knowing it can give you a deeper understanding of cyclic quadrilateral properties.

    Hint: Use Brahmagupta's formula only when the quadrilateral is known to be cyclic, as it assumes the vertices lie on a single circle!

    Angles in a Cyclic Quadrilateral

    Understanding the angles in a cyclic quadrilateral is crucial for solving various geometric problems. Cyclic quadrilaterals have distinct angle properties that set them apart from other quadrilaterals.

    Opposite Angles Property

    Opposite Angles: In a cyclic quadrilateral, the sum of each pair of opposite angles is always 180 degrees. Mathematically, if the quadrilateral is labelled as ABCD, then \(\theta_A + \theta_C = 180°\) and \(\theta_B + \theta_D = 180°\).

    Example: Suppose you have a cyclic quadrilateral ABCD. If the measure of angle A is 70°, what will be the measure of angle C?

    Since \(\theta_A + \theta_C = 180°\):

    \[\theta_C = 180° - 70° = 110°\]

    Hint: Use the property of opposite angles summing to 180° to quickly check if a quadrilateral is cyclic!

    Deep Dive: The property of opposite angles summing to 180° can be proved using the concept of inscribed angles. Consider the cyclic quadrilateral ABCD inscribed in a circle. Angles A and C subtend the same arc, hence their sum is the same as twice the inscribed angle subtended by the arc. Thus, \(\theta_A + \theta_C = 180°\). This geometric property has fascinating implications in many advanced fields of study, including trigonometry and calculus.

    Inscribed Angle Theorem

    The Inscribed Angle Theorem is another fundamental property that applies to cyclic quadrilaterals. According to this theorem, the measure of an inscribed angle is half the measure of the arc that it subtends.

    Example: In the cyclic quadrilateral ABCD, if the arc ABC subtends an angle of 100° at the centre, what is the measure of angle ADB?

    Using the Inscribed Angle Theorem:

    \[\angle ADB = \frac{100°}{2} = 50°\]

    Hint: Angles subtended by the same arc in a circle are equal! This can help in identifying congruent angles in cyclic quadrilaterals.

    Deep Dive: The Inscribed Angle Theorem can be extended to cyclic polygons. In any polygon inscribed in a circle, the angle subtended by any side at the centre is twice the angle subtended by the same side at any point on the circumcircle. This allows for several interesting relationships and can help in proving other geometric theorems involving circles.

    Area of a Cyclic Quadrilateral

    Calculating the area of a cyclic quadrilateral is an important aspect of understanding its geometry. Unlike regular quadrilaterals, a cyclic quadrilateral has special formulas for area calculation.

    Cyclic Quadrilateral Area Formula

    The area of a cyclic quadrilateral can be calculated using Brahmagupta's formula. This formula requires the lengths of all four sides of the quadrilateral.

    Brahmagupta's formula for the area \(K\) of a cyclic quadrilateral with sides of length \(a, b, c,\) and \(d\) is given by:

    \[K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\]

    where \(s\) is the semiperimeter of the quadrilateral, calculated as:

    \[s = \frac{a+b+c+d}{2}\]

    Example: Calculate the area of a cyclic quadrilateral with side lengths of 7 cm, 8 cm, 5 cm, and 10 cm.

    Using Brahmagupta's formula:

    First, find the semiperimeter \(s\):

    \[s = \frac{7+8+5+10}{2} = 15\] cm

    Next, calculate the area \(K\):

    \[K = \sqrt{(15-7)(15-8)(15-5)(15-10)}\]

    \[K = \sqrt{8 \times 7 \times 10 \times 5}\]

    \[K = \sqrt{2800} \approx 52.92\] cm²

    Hint: Make sure to check that the quadrilateral is indeed cyclic before applying Brahmagupta's formula!

    Deep Dive: Brahmagupta's formula is a specific case of a more general formula applicable to all cyclic polygons. For cyclic quadrilaterals, the formula is reminiscent of Heron's formula for the area of a triangle. It provides a neat and elegant way to deduce the area without resorting to trigonometry or calculus. This formula reflects the deep interconnection between different areas of geometry and showcases how properties of circles can simplify complex problems.

    Cyclic Quadrilateral Opposite Angles Are Supplementary Proof

    One of the most interesting properties of a cyclic quadrilateral is that its opposite angles are supplementary. This means that the sum of each pair of opposite angles is 180 degrees.

    To prove this property, let's consider a cyclic quadrilateral ABCD inscribed in a circle.

    The measure of an inscribed angle is half the measure of the arc that it subtends. Therefore:

    • The measure of angle A is half the measure of arc BCD.
    • The measure of angle C is half the measure of arc DAB.
    • Since the total measure of the circle is 360 degrees, we have:
    • \[ \text{Arc BCD} + \text{Arc DAB} = 360° \]

    Therefore:

    \[ \theta_A + \theta_C = \frac{1}{2} \times \text{Arc BCD} + \frac{1}{2} \times \text{Arc DAB} = \frac{1}{2} \times 360° = 180° \]

    This proves that the sum of opposite angles in a cyclic quadrilateral is always 180 degrees.

    Example: In a cyclic quadrilateral ABCD with angles A, B, C, and D, if \(\theta_A = 85°\), what is \(\theta_C\)?

    Using the property of supplementary opposite angles:

    \[\theta_A + \theta_C = 180°\]

    \[85° + \theta_C = 180°\]

    \[\theta_C = 180° - 85° = 95°\]

    Hint: When solving problems involving cyclic quadrilaterals, always remember the property of supplementary opposite angles!

    Deep Dive: The property of supplementary opposite angles in cyclic quadrilaterals can be applied in various geometric constructions and proofs. This characteristic is particularly useful when dealing with inscribed angles and arcs. It also has practical applications in real-life scenarios, such as designing wheels and other circular objects. Recognising this property simplifies complex problems by breaking them down into more manageable parts.

    Cyclic Quadrilaterals - Key takeaways

    • Cyclic Quadrilateral Definition: A four-sided polygon where all vertices lie on a single circle (circumcircle), with the centre called the circumcentre.
    • Properties: Opposite angles sum to 180°, perpendicular bisectors meet at the circumcentre, and Ptolemy's theorem applies: AC × BD = AB × CD + AD × BC.
    • Angles in Cyclic Quadrilateral: Sum of each pair of opposite angles is 180°, e.g., \thetaA + \thetaC = 180°.
    • Area Calculation: Brahmagupta's formula: K = sqrt((s-a)(s-b)(s-c)(s-d)), with s being the semiperimeter.
    • Supplementary Angles Proof: Use inscribed angles to show that opposite angles in a cyclic quadrilateral add up to 180°.
    Frequently Asked Questions about Cyclic Quadrilaterals
    What are the properties of a cyclic quadrilateral?
    A cyclic quadrilateral has its vertices on a common circle. Its opposite angles sum to 180 degrees. The product of the lengths of its diagonals equals the sum of the products of the lengths of opposite sides. The area can be calculated using Brahmagupta's formula.
    How can you determine if a quadrilateral is cyclic?
    A quadrilateral is cyclic if and only if the sum of its opposite angles is 180 degrees. Alternatively, if it can be inscribed in a circle, then it is cyclic.
    What is the relationship between the angles of a cyclic quadrilateral?
    In a cyclic quadrilateral, the sum of each pair of opposite angles is 180 degrees. This means that if one angle is denoted as ∠A, the opposite angle ∠C will satisfy ∠A + ∠C = 180°. The same relationship holds for the other pair of opposite angles, ∠B and ∠D.
    What is the sum of opposite angles in a cyclic quadrilateral?
    The sum of opposite angles in a cyclic quadrilateral is 180 degrees.
    What is the circumcircle of a cyclic quadrilateral?
    The circumcircle of a cyclic quadrilateral is the unique circle that passes through all four vertices of the quadrilateral.

    Test your knowledge with multiple choice flashcards

    What is a cyclic quadrilateral?

    According to the Inscribed Angle Theorem, what is the measure of an inscribed angle subtending an arc of 100° in a cyclic quadrilateral?

    What is Ptolemy's theorem for cyclic quadrilaterals?

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