Lessons 41–42 · 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
41
Test your knowledge
Textbook: pp. 114–115
GoalReview of Chapter III so far: polygons inscribed in and circumscribed about a circle, regular polygons and their circles.
New words
inscribed polygon · ichki chizilgan ko‘pburchakcircumscribed polygon · tashqi chizilgan ko‘pburchakR = a/(2sin(180°/n)) · R = a/(2sin(180°/n))r = S/p · r = S/p
Explanation
In this lesson we recall the main facts of the chapter. In an inscribed quadrilateral the opposite angles add up to 180°, while in a circumscribed quadrilateral the sums of opposite sides are equal. The circumcentre of a triangle lies where the perpendicular bisectors of the sides meet, and the incentre where the angle bisectors meet. For a circumscribed polygon S = p·r holds, where p is the semiperimeter. For a regular n-gon we use a = 2R·sin(180°/n) and r = R·cos(180°/n). Before solving a problem, draw a figure and mark clearly which circle is inscribed and which is circumscribed.
Worked examples
A circle is inscribed in a quadrilateral with consecutive sides 5, 9, 4x, 8: 5 + 4x = 9 + 8, so 4x = 12 and x = 3.
Two opposite angles of an inscribed quadrilateral are 2y and 3y + 20°: 5y + 20° = 180°, so y = 32°.
Class activity
“Two kinds of circle”: draw a triangle and find the centres of its incircle and circumcircle using bisectors and perpendicular bisectors (a compass has a sharp point, so be careful).
Practice
1
A quadrilateral with a circle inscribed in it has consecutive sides 7, 10, 12. What is the fourth side?
9
2
One angle of an inscribed quadrilateral is 75°. What is the opposite angle?
105°
3
A circumscribed polygon has semiperimeter 14 and inradius 3. What is its area?
42
4
Why are the sums of opposite sides equal in a circumscribed quadrilateral?
Because the tangent segments from each vertex to the points of tangency are equal in pairs.