Lessons 55 · 1 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
55
Final control test
Textbook: pp. 145–146
GoalSum up the Grade 9 geometry course: similarity, relations between sides and angles, circles and discs, metric relations; prepare for the final control work.
New words
similarity ratio k · o‘xshashlik koeffitsiyenti klaws of sines and cosines · sinuslar va kosinuslar teoremalaricircle and disc formulas · aylana va doira formulalarimetric relations · metrik munosabatlar
Explanation
The final control work covers four parts of the course. In similar figures lengths change k times and areas k² times. In a triangle we solve problems with the law of sines a/sinA = b/sinB = c/sinC = 2R, the law of cosines c² = a² + b² − 2ab·cosC and the area S = ½ab·sinC; the values for 30°, 45°, 60°, 120°, 150° must be known by heart. For circles: circumference C = 2πR, area of a disc S = πR², arc length l = πRn/180, sector area S = πR²n/360, and for a regular n-gon R = a/(2sin(180°/n)). Metric relations: h² = a_c·b_c, a² = c·a_c, AO·OB = CO·OD, PA² = PB·PC. Draw a figure before solving, and check the answer for size and units.
Worked examples
Sides 5 and 8 with a 60° angle between them: c² = 25 + 64 − 2·5·8·½ = 49, so c = 7.
A regular hexagon inscribed in a circle of radius 6: its area is 6·9√3 = 54√3; the disc has area 36π; the difference is 36π − 54√3.
Class activity
“Formula sheet”: write the 12 most important formulas of the course on one sheet, add a small sketch to each, and quiz a friend with it.
Practice
1
Similar triangles have ratio 3 and the smaller has area 4. What is the area of the larger?
36
2
Two sides are 5 and 8 with a 60° angle between them. What is the area?
10√3
3
a = 6, ∠A = 30°, ∠B = 45°. Find b.
6√2
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.