Cyclic Quadrilateral Practice Problems

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.

What is a Cyclic Quadrilateral?

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.

Properties of Cyclic Quadrilateral

Common properties of cyclic quadrilateral are:

Radius of Circumcircle

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:

R = \frac{\sqrt{(s-a)(s-b)(s-c)(s-d)}}{4K}

Where s = \frac{a + b + c + d}{2} is the semi perimeter and K is the area of the quadrilateral.

Diagonals

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.

Formulas Related to Cyclic Quadrilateral

Here’s a table summarizing key formulas related to the cyclic quadrilaterals:

Important Related Formulas

Sum of Opposite Angles: The sum of opposite angles = 180^\circ

Ptolemy’s Theorem: For a cyclic quadrilateral ABCD Ptolemy’s Theorem states:

AC \cdot BD = AB \cdot CD + BC \cdot DA

Area of a Cyclic Quadrilateral: The area K can be calculated using Brahmagupta’s formula:

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

Where,

s = \frac{a + b + c + d}{2} is the semi perimeter and a, b, c, d are the side lengths.

Special Case - Rectangle: For a rectangle the area is given by:

Area = \text{length} \times \text{width}

Special Case - Rhombus: For a rhombus the area can also be found using:

Area = \frac{1}{2} \times d_1 \times d_2

Where d₁ and d₂ are the lengths of the diagonals.

Solved Examples of Cyclic Quadrilateral

Examples 1: In a cyclic quadrilateral if the measures of two opposite angles are 70^\circ and 110^\circ find the measures of other two angles.

Solution:

The sum of opposite angles in a cyclic quadrilateral is 180^\circ. Thus,

Measure of angle 3 = 180^\circ - 70^\circ = 110^\circ

Measure of angle 4 = 180^\circ - 110^\circ = 70^\circ

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:

Area = \sqrt{(13 - 5)(13 - 7)(13 - 8)(13 - 6)} = \sqrt{8 \cdot 6 \cdot 5 \cdot 7} = \sqrt{1680} \approx 41

Examples 3: The Prove that if a quadrilateral is cyclic then the sum of its opposite angles is 180^\circ.

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 180^\circ.

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:

AC \cdot BD = AB \cdot CD + AD \cdot BC

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 d_1 and d_2 with d_1 = 2d_2.

Solution:

The area of a cyclic quadrilateral with perpendicular diagonals is:

Area = \frac{1}{2} \times d_1 \times d_2

Substitute d_1 = 2d_2:

Area = \frac{1}{2} \times 2d_2 \times d_2 = d_2^2

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:

2x + 3x + 4x + 5x = 14x = 72 \implies x = \frac{72}{14} = 5.14

Thus, the side lengths are 2 \cdot 5.14, 3 \cdot 5.14, 4 \cdot 5.14, 5 \cdot 5.14.

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:

s = \frac{10 + 15 + x + x}{2} = 20 \implies 25 + 2x = 40 \implies x = 7.5

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:

s = \frac{6 + 8 + 10 + 12}{2} = 18

Area = \sqrt{(18 - 6)(18 - 8)(18 - 10)(18 - 12)} = \sqrt{12 \cdot 10 \cdot 8 \cdot 6} = \sqrt{5760} \approx 75.9

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:

Area = \frac{1}{2} \times 7 \times 24 = 84

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:

\angle A + \angle C = 180^\circ \text{ and } \angle B + \angle D = 180^\circ

By the inscribed angle theorem this confirms that the quadrilateral is cyclic.

Practice Questions on Cyclic Quadrilateral

Questions 1: Find the measure of the angles of the cyclic quadrilateral if one angle is 50^o and another is 120^o.

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.

Conclusion

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.

Read More,

• What is Cyclic Quadrilateral
• Quadrilaterals Practice Questions