☰ Contents · Geometry

Inscribed and circumscribed polygons

Lessons 36–37 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
36

A polygon inscribed in a circle

Textbook: pp. 104–105
GoalKnow the inscribed polygon; decide which polygons have a circumscribed circle and where its centre lies.
New words
inscribed polygon · ichki chizilgan ko‘pburchakcircumscribed circle · tashqi chizilgan aylanaperpendicular 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 it. A circle can be circumscribed about any triangle; its centre lies at the intersection of the perpendicular bisectors of the sides, since the centre is equally far from the vertices. For polygons with more sides this is not always possible: a quadrilateral has a circumscribed circle if and only if its opposite angles add up to 180°. So a rectangle and an isosceles trapezoid have one, but a parallelogram that is not a rectangle does not. In a right triangle the hypotenuse is a diameter of the circumscribed circle.

Worked examples
A right triangle with legs 6 and 8: the hypotenuse 10 is a diameter, so R = 5.
In an inscribed ABCD with ∠A = 100° and ∠B = 70°: ∠C = 180° − 100° = 80° and ∠D = 180° − 70° = 110°.
Class activity

“Find the centre”: draw a triangle on paper, construct the perpendicular bisectors of two sides with a compass and ruler, and check that the distances from their intersection to the vertices are equal.

Practice
1
What is the radius of the circle circumscribed about a 6 × 8 rectangle?
2
In a quadrilateral inscribed in a circle, ∠A = 85°. What is the opposite angle ∠C (degrees)?
3
A right triangle has hypotenuse 26 cm. What is the radius of its circumscribed circle?
4
Why can no circle be circumscribed about a parallelogram that is not a rectangle?