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
37
A polygon circumscribed about a circle
Textbook: pp. 106–107
GoalKnow the circumscribed polygon; the condition for a quadrilateral to have an inscribed circle, and apply S = pr.
New words
circumscribed polygon · tashqi chizilgan ko‘pburchakinscribed circle (incircle) · ichki chizilgan aylanasemi-perimeter · yarim perimetrangle bisector · bissektrisa
Explanation
If all sides of a polygon are tangent to a circle, the polygon is circumscribed about the circle and the circle is inscribed in the polygon. The centre lies at the intersection of the angle bisectors, since it is equally far from the sides. Every triangle has an incircle. A quadrilateral has an inscribed circle if and only if the sums of opposite sides are equal: AB + CD = BC + AD. So a rhombus and a square have an incircle, but a non-square rectangle does not. Theorem: for a polygon circumscribed about a circle of radius r, the area is S = pr, where p is the semi-perimeter. The polygon splits into triangles from the centre, each with height r.
Worked examples
In ABCD circumscribed about a circle with AB = 5, CD = 7, BC = 9: 5 + 7 = 9 + AD, so AD = 3.
A polygon with perimeter 30 and incircle radius 4: p = 15 and S = pr = 15 · 4 = 60.
Class activity
“Bisector centre”: draw a triangle, construct two angle bisectors, and measure the perpendiculars from their intersection to the sides to check they are equal.
Practice
1
A right triangle has legs 6 and 8 and hypotenuse 10. Using r = (a + b − c)/2, find r.
2
2
Three consecutive sides of a quadrilateral circumscribed about a circle are 4, 6 and 9. What is the fourth side?
7
3
A circumscribed polygon has area 84 and perimeter 42. What is the radius of the circle?
4
4
Why can a non-square rectangle not have an inscribed circle?
The opposite sides give a + a and b + b, which are equal only when a = b.