Lessons 18 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
18
Practice and applications: planimetry
Textbook, Part 1: pp. 108–111
GoalReview the main facts of planimetry (angles, triangle, quadrilaterals, circle) and apply them to calculation and justification problems.
New words
alternate interior angles · ichki almashinuvchi burchaklarmidline · o‘rta chiziqchord · vatarinscribed circle · ichki chizilgan aylana
Explanation
Vertical angles formed by two intersecting lines are equal and adjacent angles sum to 180°. When a transversal cuts two parallel lines, alternate interior angles and corresponding angles are equal and co-interior angles sum to 180°. The angles of a triangle sum to 180°, an exterior angle equals the sum of the two interior angles not adjacent to it; a convex n-gon has interior angle sum (n − 2) · 180°. The midline of a trapezoid equals half the sum of the bases, the area of a rhombus is half the product of its diagonals (S = d₁d₂/2), and its diagonals are perpendicular and bisect each other. The circumference is 2πR, the area of a disc is πR²; the radius of the circle inscribed in a square is half the side; if the distance from the centre to a chord is d, half the chord equals √(R² − d²), since the perpendicular from the centre bisects the chord. When solving, first draw a figure, mark the data and pick the matching fact.
Worked examples
A trapezoid has area 72 cm² and height 8 cm. Since S = (a + b)/2 · h, the midline is m = (a + b)/2 = S/h = 72/8 = 9 cm. If one base is 12 cm, the other is 2 · 9 − 12 = 6 cm.
A chord is drawn in a circle of radius 13 cm at a distance 5 cm from the centre. Half the chord is √(13² − 5²) = √144 = 12 cm, so the chord is 24 cm. Half the chord, d and R form a right triangle (Pythagorean theorem).
Class activity
Review relay: the class splits into two teams; six short problems (angle, trapezoid, rhombus, circle) are written on the board. Team members take turns solving one each, and the previous member checks the answer.
Practice
1
How is the sum of the interior angles of a convex n-gon found?
(n − 2) · 180°
2
The interior angle sum of a convex polygon is 720° more than that of a quadrilateral. How many sides does it have?
8
3
A rhombus has side 13 cm and one diagonal 24 cm. Find the other diagonal and the area.
The half-diagonals are 12 and √(13² − 12²) = 5, so the other diagonal is 10 cm; S = 24 · 10/2 = 120 cm².
4
Why does the perpendicular from the centre to a chord bisect the chord?
Joining the chord’s ends to the centre gives an isosceles triangle (two sides are radii). In an isosceles triangle the altitude to the base is also the median.