Facts About Cyclic Quadrilateral

Cyclic quadrilateral is a special kind of quadrilateral inscribed in a circle. Inscribe means that the vertices of quadrilateral are on the circumference of the circle. A square, rectangle, isosceles trapezoid are some of cyclic quadrilaterals. Below are the some facts about this special figure.

Cyclic Quadrilateral

1. The opposite angle of cyclic quadrilateral is supplementary.

\angle{A}+\angle{C}=180°

\angle{B}+\angle{D}=180°

2. Ptolemy’s Theorem. This theorem is a special case of Ptolemy’s Inequality. He formulated the following equation relating the sides and diagonals.

d_1d_2=ac+bd

3. Diagonals as Cords. Using the basic theorem in geometry, we can relate the diagonals with respect to their intersections as follows.

AO(OC)=OD(OB)

4. Brahmagupta’s Formula. Brahmagupta of India derived a formula to find the area of cyclic quadrilateral.

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

Where S is the semiperimeter

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

Use the facts above to solve for the following problems.

Problem 1:  In a cyclic quadrilateral ABCD, If \angle{A}=105°. Find the measure of \angle{C}

Solution:

Using the first fact, opposite angles are supplementary.

\angle{A}+\angle{C}=180°

105°+\angle{C}=180°

\angle{C}=(180-105)°

\angle{C}=75°

Problem 2: In cyclic quadrilateral ABCD, \angle{C}=\angle{A}=90°. If \overline{BD}=5, and two opposite sides have length 2 and 3. What is the area of quadrilateral?

 

Solution:

Draft the quadrilateral as follows,

cyclic quadrilateralSince \Delta BCD and \Delta ABD are right triangle and \overline{BD} is the hypotenuse of both triangles, we can find sides \overline{AD}  and   \overline{BC}  by Pythagorean Theorem.

\overline{BD}^2=\overline{BC}^2+\overline{CD}^2

5^2=\overline{BC}^2+3^2

\overline{BC}^2=25-9=16

\overline{BC}=4

\overline{BD}^2=\overline{AB}^2+\overline{AD}^2

5^2=2^2+\overline{AD}^2

\overline{AD}^2=25-4=21

\overline{AD}=\sqrt{21}

The area can be found using Brahmagupta’s formula or using the fact that both triangles are right triangle.

Area_{quadrilateral}=Area_{\Delta BCD}+Area_{\Delta ABD}

Area_{quadrilateral}=\displaystyle\frac{\overline{BC}\cdot \overline{CD}}{2}+\displaystyle\frac{\overline{AB}\cdot \overline{AD}}{2}

Area_{quadrilateral}=\displaystyle\frac{1}{2}(3\cdot 4+2\cdot \sqrt{21})

Area_{quadrilateral}=6+\sqrt{21} sq. units

 

 

 

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

Latest posts by Dan (see all)

You may also like...

9 Responses

  1. Harold says:

    I like what you dudes are now up to. This sort of cool work and coverage! Keep up the great work guys, I’ve included you our blogroll.

  2. Great website you have here but I was curious about if you knew of any forums that cover the same topics talked about in this article? I’d really love to be a part of community where I can get comments from other knowledgeable people that share the same interest. If you have any suggestions, please let me know. Bless you!

  3. this site says:

    GNM54x This is very interesting, You are a very skilled blogger. I have joined your rss feed and look forward to seeking more of your wonderful post. Also, I ave shared your web site in my social networks!

  4. suba me says:

    wIRAKC Really informative article. Much obliged.

  5. 801509 751757This internet website is often a walk-through rather than the details you wanted about it and didnt know who ought to. Glimpse here, and youll definitely discover it. 378161

  6. viagra says:

    844007 188477These kinds of Search marketing boxes normally realistic, healthy and balanced as a result receive just about every customer service necessary for some product. Link Building Services 329182

  7. Hi,I log on to your new stuff named “Facts About Cyclic Quadrilateral – Techie Math Teacher” like every week.Your story-telling style is witty, keep it up! And you can look our website about proxy server list.

  8. lace frontal says:

    lace frontal https://www.youtube.com/watch?v=ny8rUpI_98I're genuinely cute and fashion. I purchased the one particular and love them.

  9. Way cool! Some extremely valid points! I appreciate you penning this write-up plus the rest of the site is very good.

Leave a Reply

Your email address will not be published.