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VOOZH | about |
A Cyclic quadrilateral is a four-sided figure that lies entirely on the circumference of one circle. This specific feature produces several interesting geometric theorems and properties useful in solving varied mathematical problems. Thus, learners must comprehend cyclic quadrilaterals by broadening their problem-solving capabilities in geometry and gaining insights into other advanced ideas.
A cyclic quadrilateral is a four-sided polygon where all its vertices lie on the circumference of the single circle. This circle is known as the circumcircle or circumscribing circle of the quadrilateral. In simpler terms, a cyclic quadrilateral is one where there exists a circle that passes through the all four vertices of the quadrilateral.
Common properties of cyclic quadrilateral are:
Property | Description |
|---|---|
Opposite Angles | The sum of the measures of opposite angles is 180o |
Circumcircle | All four vertices lie on the single circle. |
The exterior angle of the cyclic quadrilateral is equal to the interior opposite angle. | |
Ptolemy’s Theorem | In a cyclic quadrilateral, the sum of the products of its two pairs of the opposite sides is equal to product of the diagonals. |
Angle at the Circumference | The Angles subtended by the same chord are equal. |
For a cyclic quadrilateral with the sides a, b, c and d the radius R of the circumcircle can be computed using the following formula:
Where is the semi perimeter and K is the area of the quadrilateral.
For a cyclic quadrilateral, the relation between the diagonals e and f and the sides is given by:
e2 + f2 = a2 + b2+ c2 + d2
Where e and f are the lengths of the diagonals and a, b, c and d are the sides of the quadrilateral.
Here’s a table summarizing key formulas related to the cyclic quadrilaterals:
Formula | Description |
|---|---|
∠A + ∠C = 180° | The Sum of opposite angles of a cyclic quadrilateral. |
∠B + ∠D = 180° | Another pair of the opposite angles. |
AC ⋅ BD = AB ⋅ CD + BC ⋅ AD | The Ptolemy’s Theorem for the cyclic quadrilaterals. |
Sum of Opposite Angles: The sum of opposite
Ptolemy’s Theorem: For a cyclic quadrilateral ABCD Ptolemy’s Theorem states:
Area of a Cyclic Quadrilateral: The area K can be calculated using Brahmagupta’s formula:
Where,
is the semi perimeter and a, b, c, d are the side lengths.
Special Case - Rectangle: For a rectangle the area is given by:
Special Case - Rhombus: For a rhombus the area can also be found using:
Where d1 and d2 are the lengths of the diagonals.
Examples 1: In a cyclic quadrilateral if the measures of two opposite angles are and find the measures of other two angles.
Solution:
The sum of opposite angles in a cyclic quadrilateral is . Thus,
Measure of angle 3 =
Measure of angle 4 =
Examples 2: In a cyclic quadrilateral ABCD if AB = 5, BC = 7, CD = 8 and DA = 6 find the area of the quadrilateral if the semi perimeter is 13.
Solution:
Using Brahmagupta’s formula:
Examples 3: The Prove that if a quadrilateral is cyclic then the sum of its opposite angles is .
Solution:
By definition a quadrilateral is cyclic if all its vertices lie on a circle. The inscribed angle theorem states that an angle subtended by a chord of a circle is half of the angle subtended by the chord on the other side. Hence, the sum of the measures of the opposite angles in the cyclic quadrilateral is .
Examples 4: The Calculate the length of the diagonal AC in a cyclic quadrilateral ABCD where AB = 3, BC = 4, CD = 5, DA = 6 using the Ptolemy’s Theorem.
Solution:
Ptolemy’s Theorem:
Since AC and BD are unknown solve the equation if BD is known or use another method to the find BD first.
Examples 5: In a cyclic quadrilateral if the diagonals intersect at right angles and one diagonal is twice as long as the other find the area of the quadrilateral if the lengths of the diagonals are and with .
Solution:
The area of a cyclic quadrilateral with perpendicular diagonals is:
Substitute :
Examples 6: Given a cyclic quadrilateral where the sides are in the ratio 2:3:4:5 find the length of the each side if the perimeter is 72.
Solution:
Let the sides be 2x,3x,4x,5x. The perimeter is:
Thus, the side lengths are .
Examples 7: If one of the sides of the cyclic quadrilateral is 10 units and opposite side is 15 units and other two sides are equal find the lengths of the two equal sides if the semi perimeter is 20.
Solution:
Let the lengths of the two equal sides be x. The semi perimeter s is:
Examples 8: Find the area of the cyclic quadrilateral with sides 6, 8, 10, 12 using the Brahmagupta’s formula.
Solution:
Calculate the semi perimeter:
Examples 9: In a cyclic quadrilateral, if the lengths of the diagonals are 7 and 24 find the area if the diagonals are perpendicular to the each other.
Solution:
The area of the cyclic quadrilateral is:
Examples 10: Prove that if the opposite angles of the quadrilateral are supplementary then the quadrilateral is cyclic.
Solution:
Assume the quadrilateral ABCD has opposite angles supplementary. Then:
By the inscribed angle theorem this confirms that the quadrilateral is cyclic.
Questions 1: Find the measure of the angles of the cyclic quadrilateral if one angle is 50o and another is 120o.
Questions 2: Calculate the area of the cyclic quadrilateral with sides 5, 12, 13 and 14 units.
Questions 3: Given a cyclic quadrilateral with the diagonals 10 and 15 units find the area if they intersect at right angles.
Questions 4: If the sides of a cyclic quadrilateral are in the ratio 3:4:5:6 and the perimeter is 36 units find the length of the each side.
Questions 5: Prove that the diagonals of the cyclic quadrilateral are not necessarily perpendicular.
Questions 6: Find the area of a cyclic quadrilateral with the sides 7, 24, 25 and 30 units.
Questions 7: Determine the length of the diagonals in the cyclic quadrilateral where the sides are 8, 15, 20 and 25 units.
Questions 8: If the area of a cyclic quadrilateral is 48 square units and one diagonal is 10 units find the length of the other diagonal if they are perpendicular.
Questions 9: Calculate the missing side lengths of the cyclic quadrilateral with the given semi perimeter of 18 and known sides 6 and 8.
Questions 10: Prove that if the opposite angles of the quadrilateral are not supplementary then the quadrilateral is not cyclic.
The Cyclic quadrilaterals possess unique properties that make them a fascinating subject in the geometry. By understanding and applying the key theorems such as the sum of the opposite angles and Ptolemy's theorem we can solve a wide range of the problems involving the cyclic quadrilaterals. The Mastering these concepts will enhance the problem-solving skills and deepen the understanding of the geometric relationships.
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