Quadrilaterals in a Circle – Explanation & Examples

Quadrilaterals in a CircleWe have studied that a quadrilateral is a 4 – sided polygon with 4 angles and 4 vertices. For more details, you can consult the article “Quadrilaterals” in the “Polygon” section.

In geometry exams, examiners make the questions complex by inscribing a figure inside another figure and ask you to find the missing angle, length, or area. One example from the previous article shows how an inscribed triangle inside a circle makes two chords and follows certain theorems.

This article will discuss what a quadrilateral inscribed in a circle is and the inscribed quadrilateral theorem.

What is a Quadrilateral Inscribed in a Circle?

In geometry, a quadrilateral inscribed in a circle, also known as a cyclic quadrilateral or chordal quadrilateral, is a quadrilateral with four vertices on the circumference of a circle. In a quadrilateral inscribed circle, the four sides of the quadrilateral are the chords of the circle.

What is a Quadrilateral Inscribed in a Circle

In the above illustration, the four vertices of the quadrilateral ABCD lie on the circle’s circumference. In this case, the diagram above is called a quadrilateral inscribed in a circle.

Inscribed Quadrilateral Theorem

There are two theorems about a cyclic quadrilateral. Let’s take a look.

Theorem 1

The first theorem about a cyclic quadrilateral state that:

The opposite angles in a cyclic quadrilateral are supplementary. i.e., the sum of the opposite angles is equal to 180˚.

Consider the diagram below.

If a, b, c, and d are the inscribed quadrilateral’s internal angles, then

a + b = 180˚ and c + d = 180˚.

The opposite angles in a cyclic quadrilateral are supplementary

Let’s prove that;

  • a + b = 180˚.

Join the vertices of the quadrilateral to the center of the circle.

THEOPP2

Recall the inscribed angle theorem (the central angle = 2 x inscribed angle).

COD = 2∠CBD

COD = 2b

Similarly, by intercepted arc theorem,

COD = 2 CAD

COD = 2a

COD + reflex ∠COD = 360o

2a + 2b = 360o

2(a + b) =360o

By dividing both sides by 2, we get

a + b = 180o.

Hence proved!

Theorem 2

The second theorem about cyclic quadrilaterals states that:

The product of the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the product of its two pairs of opposite sides.

Consider the following diagram, where a, b, c, and d are the sides of the cyclic quadrilateral and D1 and D2 are the quadrilateral diagonals.

THEPRO1

In the above illustration,

(a * c) + (b * d) = (D1 * D2)

Properties of a quadrilateral inscribed in a circle

There exist several interesting properties about a cyclic quadrilateral.

  • All the four vertices of a quadrilateral inscribed in a circle lie on the circumference of the circle.
  • The sum of two opposite angles in a cyclic quadrilateral is equal to 180 degrees (supplementary angles)
  • The measure of an exterior angle is equal to the measure of the opposite interior angle.
  • The product of the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the product of its two pairs of opposite sides.
  • The perpendicular bisectors of the four sides of the inscribed quadrilateral intersect at the center O.
  • The area of a quadrilateral inscribed in a circle is given by Bret Schneider’s formula as:

Area = √[s(s-a) (s-b) (s – c) (s – c)]

where a, b, c, and d are the side lengths of the quadrilateral.

s = Semi perimeter of the quadrilateral = 0.5(a + b + c + d)

Let’s get an insight into the theorem by solving a few example problems.

Example 1

Find the measure of the missing angles x and y in the diagram below.

opposite angles are supplementary

Solution

x = 80 o (the exterior angle = the opposite interior angle).

y + 70 o = 180 o (opposite angles are supplementary).

Subtract 70 o on both sides.

y = 110o

Therefore, the measure of angles x and y are 80o and 110o, respectively.

Example 2

Find the measure of angle ∠QPS in the cyclic quadrilateral shown below.

measure of angle in the cyclic quadrilateral

Solution

QPS is the opposite angle of ∠SRQ.

According to the inscribed quadrilateral theorem,

QPS + ∠SRQ = 180o (Supplementary angles)

QPS + 60o = 180o

Subtract 60o on both sides.

QPS = 120 o

So, the measure of angle ∠QPS is 120o.

Example 3

Find the measure of all the angles of the following cyclic quadrilateral.

measure of all missing angles of the cyclic quadrilateral

Solution

Sum of opposite angles = 180 o

(y + 2) o + (y – 2) o = 180 o

Simplify.

y + 2 + y – 2 =180 o

2y = 180 o

Divide by 2 on both sides to get,

y = 90 o

On substitution,

(y + 2) o ⇒ 92 o

(y – 2) o ⇒ 88 o

Similarly,

(3x – 2) o = (7x + 2) o

3x – 2 + 7x + 2 = 180 o

10x =180 o

Divide by 10 on both sides,

x = 18 o

Substitute.

(3x – 2) o ⇒ 52 o

(7x + 2) o ⇒ 128o

 

Practice Questions

1. True or False:  All polygons can be inscribed in a circle.

2. Fill in the blank: Inscribed quadrilaterals are also called __________.

3. Fill in the blank: A quadrilateral is inscribed in a circle if and only if the opposite angles are __________.

4. What is the value of $x$ using the diagram shown below?

finding angles from quadrilaterals in a circle

5. What is the value of $y$ using the diagram shown below?

finding angles from quadrilaterals in a circle


 

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