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), |AB| = |AE| + |Exist|, From the other side, | Effigy 1. To the Problem ane |
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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