How to Draw a Quadrilateral in a Circle

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Quadrilateral circumscribed about a circle

In this lesson y'all will larn that a quadrilateral confining about a circle has a specila belongings - the sums of the measures of its reverse sides are equal.
The theoretical base for solving these problems is the lesson  Tangent segments to a circle from a indicate outside the circle  nether the topicCircles and their properties
of the sectionGeometry  in this site.

Problem 1

If a quadrilateral is circumscribed about a circle,  then the sums of its opposite sides are equal.

Solution LetABCD  be a quadrilateral circumscribed about a circle  (Figure 1), and permitEastward,F,Grand  andH  be the tangent points of the segmentsAB, BC,CD  andAd  respectively  (the sides of the quadrilateral) and the circle.  We accept |AB| = |AE| + |Exist|, |BC| = |BF| + |CF|, |CD| = |CG| + |DG|, |Advert| = |AH| + |DH|. From the other side,

Effigy 1.  To the Problem ane

|AE| = |AH|,  |BE| = |BF|,  |CF| = |CG|  and  |AH| = |DH|

in accordance with the lesson  Tangent segments to a circle from a point outside the circle  nether the topicCircles and their properties  of the section Geometry
in this site.  Therefore,

|AB| + |CD| = |AE| + |BE| + |CG| + |DG| = |AH| + |BF| + |CF| + |AB| + |DH| = (|AH| + |DH|) + (|BF| + |CF|) = |AD| + |BC|.

Thus  |AB| + |CD| = |AD| + |BC|.  It is what has to be proved.  The solution is completed.

Example 1

Find the measure of the quaternary side of a quadrilateral circumscribed nearly a circle,  if 3 other sides have the measures of  5 cm,  half dozen cm  and  iv cm  listed consecutively.

Solution

Letx  be the measure of the fourth side of our quadrilateral.  Since the quadrilateral is circumscribed about a circumvolve,  the sums of the measures of its contrary sides are
equal  in accordance with theProblem 1  above.  Thus you can write the equation

5 + 4 = six + x.

From this equation,x = v + 4 - 6 = iii cm.

Answer. The fourth side of the quadrilateral is of  3 cm  long.

Instance two

A trapezoid is circumscribed nigh a circle.  Detect the measure of the mid-segment of a trapezoid,  if its lateral sides are of  5 cm  and  seven cm  long.

Solution

Since the trapezoid is circumscribed nigh a circle,  the sums of the measures of its reverse sides are equal in accord with theProblem i  to a higher place.  Thus the sum
of the measures of its bases is equal to the sum of the measures of its lateral sides,  i.e.  5 + seven = 12 cm.
The mid-segment of a trapezoid has the measure out half the sum of the measures of its bases  (see the lesson  Trapezoids and their mid-lines  under the topicPolygons
of the departmentGeometry  in this site.  So,  the mid-segment of our trapezoid has the measure of = 6 cm.
Answer.  The mid-segment of the trapezoid is of  6 cm  long.

Example 3

The sides of a quadrilateral are of  v cm,  6 cm,  seven sm  and  8 cm  long  listed consecutively.  Prove that this quadrilateral is non circumscribed about a circle.

Solution

If a quadrilateral is circumscribed about a circle,  so the sums of its contrary sides are equal.  In our case the sums of the opposite sides are of  5 + seven = 12 cm
and  6 + 8 = 14 cm.  Since the sums are not equal,  the quadrilateral is not circumscribed about a circle.

If you want to navigate among my other lessons on Polygons in this site, then apply this list of links:
- Sum of interior angles of a polygon,
- Quadrilateral inscribed in a circle
- Regular polygons,
- The side length of a regular polygon via the radius of the circumscribed circle,
- The side length of a regular polygon via the radius of the inscribed circle,
- Miscellaneous issues on polygons  and
- Backdrop OF POLYGONS
under the topicPolygons  of the sectionGeometry,  and
- Solved problems on interior angles of a polygon  and
- Solved bug on the side length of a regular polygon
under the topicGeometry  of the sectionWord issues.

If you desire to navigate among my lessons on circles, their chords, secant and tangent lines, then employ these links
- A circle, its chords, tangent and secant lines - the major definitions,
- The longer is the chord the larger its cardinal angle is,
- The chords of a circle and the radii perpendicular to the chords,
- A tangent line to a circumvolve is perpendicular to the radius drawn to the tangent point,
- An inscribed angle in a circle,
- Ii parallel secants to a circumvolve cut off congruent arcs,
- The angle betwixt ii chords intersecting inside a circle,
- The angle betwixt two secants intersecting exterior a circle,
- The angle betwixt a chord and a tangent line to a circle,
- Tangent segments to a circle from a indicate outside the circle,
- The antipodal theorem on inscribed angles,
- The parts of chords that intersect inside a circle,
- Metric relations for secants intersecting outside a circle  and
- Metric relations for a tangent and a secant lines released from a indicate outside a circle
- Quadrilateral inscribed in a circle
- PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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