Lessons 38–39 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
39
The circumscribed circle
Textbook: pp. 121–123
GoalKnow the properties of a circumcircle and of an inscribed polygon.
New words
circumcircle · tashqi chizilgan aylanainscribed polygon · ichki chizilgan ko‘pburchakperpendicular bisector · o‘rta perpendikulyaropposite angles · qarama-qarshi burchaklar
Explanation
If all vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon. Every triangle has exactly one circumcircle; its centre is the intersection point of the perpendicular bisectors of the sides, since every point of a perpendicular bisector is equally far from the segment’s ends. Opposite angles of an inscribed quadrilateral add up to 180°, because they are measured by half the sum of the two arcs, and those arcs make 360°. The converse is true too: a convex quadrilateral whose opposite angles add up to 180° has a circumcircle. In a right triangle the circumcentre is the midpoint of the hypotenuse.
Worked examples
In an inscribed quadrilateral ∠A = 80° and ∠B = 110°: ∠C = 100° and ∠D = 70°.
A triangle has angles 50°, 60°, 70°. From the circumcentre its sides are seen at 100°, 120°, 140°.
Class activity
“Three points, one circle”: mark three points, draw two perpendicular bisectors and, with an adult’s help, draw the circle with a compass from their intersection.
Practice
1
In an inscribed quadrilateral ∠A = 95°. How many degrees is ∠C?
85
2
One angle of a triangle is 65°. At how many degrees is the side opposite it seen from the circumcentre?
130
3
What is the radius of the circle circumscribed about a right triangle with legs 6 and 8?
5
4
Why is the circumcentre of a right triangle the midpoint of the hypotenuse?
The right angle subtends a diameter, so the hypotenuse is a diameter.