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1O Geometry I - Rectilineor Figures C. Triangle inequality The sum of the lengths of any two sides of a triangle is greater than the length the e.g. third side. This relation is known as of triangle inequalitl'. In AABC, a+b> c,and b+c>a,and c+a>b. \&'e can apply the above result to determine rvhether three line segments with given lengths can form a triangle. rc.7 Quadrilaterals A. Properties of parallelograms A parallelogram is a quadrilateral with two pairs of parallel opposite sides. If ABCD is a parallelogram, then AB=DC and AD=BC lA= lC (Opp.Sides of//gram) (opp. Zs of Ar-------l'-- and ll lB = lD (iii)AO=OC and BO=OD gram) ll gram) D ハ ハ D (diags. of ー 0 ー B. Conditions for parallelogramr (i) If PQ = SR and P$ = QR, then PQRS is a parallelogram. (opp. sides equal) (iii) If PO = OR and QO = (ii) If ZP= lRand lQ= ZS,then PQRS is a parallelogram. (opp. Zs equal) PQRS is a parallelogram. alld PS〃 QR,then PQRS is a paraudogram. (diags. bisect each other) (Opp.Sides equal and//) OS, then (市 )IfPS=QR 0 Note: The proofs related to parallelograms belong to the non-foundation part of the syllabus. ang e nequal性 y三 角不等式 paral e ogram平 イ 子四邊形 163 L Geometry I - Rectilineor Figures c. Properties of rhombuses, rectangtes, squares, trapeziums and kites (a) Rhombuses A (i) is a parallelogram with four equal sides It has all properties of a parallelogram. (ii) All its sides are equal in length. (iii) Its diagonals are perpendicular to each other. (iv) Its interior angles are bisected by the diagonals. (Abbreviation: property of rhombus) (b) Rectangles A rectangle is a parallelogram with four interior right angles. (i) It has all properties of a parallelogram. (i0 All the interior angles are right angles. (iii) Its diagonals are equal in length. (rv) Its diagonals bisect each other into four equal parts. (Abbreviation: property of rectangle) (C) Squares A (i) It has all properties of a rectangle. (ii) It has all properties of a rhombus. is a rectangle with four equal sides. { A square is both a rhombus and a rectangle. (iii) Each angle between a diagonal and a side is 45o. (Abbreviation: property of square) (d) Trapeziums A trapezium is a quadrilateral with only one pair of paraliel opposite trapezium are: (i) sides. Two special kinds A trapezium with one of its sides perpendicular to the t.,yo bases is called a right-angled trapezium. . (ii) A trapezium with non-parallel sides of equal length following properties: ∠P=∠ S and∠ Q=∠ R P rhombus麦 形 " 1要 PR=QS s rectang e長 方形 isosceles trapezium等 ′‐ called an isosceles trapezium. It has the 梯形 square E)1fu lght― angttd trapezium直 角梯形 tO Geometry I - Rectilineor (e)Kites A is a quadrilateral with two pairs ofequal attacent sides. Consider kite ABCD,where AB=AD and CB=CD.AC intersects BD at O. ￨￨ It has the follo、 ving properties: (i)∠ ABC=∠ ADC (ii)BOth∠ BAD and∠ BCD are bisected bv AC. (ili)BO=D0 (市 )AC tt BD Ｄ Ｃ 10.8 Mid-point Theorern and lntercept Theorenn A. Mid-point theorem In AABC, 1f AM = MB andAN = NC, then (i) MN il BC, (i0 I MN =-BC. 2 (mid-pt. theorem) B. Intercept theorem then DE=EF. In△ ABC,ifAE=EB and EF〃 BC,then AF=FC. (intercept theorelln) (intercept theorem) If AD//BE//CF and AB=BC, S Of 0 s the Example饉 : In the igure,A3//CD,∠ 34K=30° ,∠ AKD=130° 狙 d 1警 ‐ '摯 警 鞭 響 諄 撃■ ∠ ´te鳶 形 cDK=ノ .Find夕 mld po nttheorem中 黙定理 . ‐ 7定 理 ntercept theorem載 ‐ ー