Discovering Geometry and Measurement Concepts in Grade 9 Mathematics

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Explore the fundamentals of geometry and measurement in grade 9 math, covering topics such as regular polygons, congruence and similarity of triangles, construction of similar figures, trigonometric ratios application, circle properties, and problem-solving related to triangles and parallelograms. Understand polygon definitions, convex and concave polygons, diagonal concepts, interior and exterior angles, regular polygons, apothem distances, and formulas for sides, apothem, perimeter, and area. Delve into the congruency of triangles through the concept of side-side-side (SSS) congruence.

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GRADE 9 MATHS GRADE 9 MATHS

Geometry and Measurement OBJECTIVES OF THE LESSON After completing this unit you should be able to Know the basic concepts of regular polygons Prove congruence and similarity of triangles Construct similar figures Apply the concepts of trigonometric ratios to solve Problems in practical situations Known facts about circles Solve problems on areas of triangles and parallel grams

DEFINITION DEFINITION A polygon is a simple closed curve formed by the union of three or more line segments , no two of which in succession are collinear . The line segments are called the sides of the polygon and the end points of the sides are called the vertices .

Convex polygon if each interior angle less than 1800 Concave polygon (non-convex) if at least one of its interior angel is greater than 1800. A diagonal of a polygon is a line segment that joins any two of its nonconsecutive vertices. The sum of all interior angels of an n- sided polygon is given by the formula The sum of all exterior angel of a polygon is given by The polygon with all sides and angles are equal is called a regular polygon S= (n-2) 1800 S = 3600

The polygon with all sides and angles are equal is called a regular polygon Interior angle of each regular polygon is obtained by (? 2)1800 ? Each exterior angel of a regular polygon of n- sided is 3600 ? Each central angel of a regular n- sided polygon is 3600 ? The apothem is the distance from the center of regular polygon to a side of the polygon.

Formula for the length of Sides s, S= 2r sin 1800 ? Apothem a, A= r cos1800 ? Perimeter p, and P= 2 nr sin 1800 ? 2 n ?2sin 3600 Area A of a, A= 1 2?? ?? ( A= 1 ?) Regular polygon is given by Regular polygon is given by

Congruency of triangles Congruency of triangles Two triangles are congruent, if the following corresponding corresponding of the triangles are congruent. Three sides (sss)

Two angles and included Two angles and included sided (ASA) sided (ASA) two angles and included two angles and included side (ASA) side (ASA)

Two sides and Two sides and TWO SIDED AND INCLUDED TWO SIDED AND INCLUDED angels angels (SAS) (SAS)SAS) SAS)

A A Aright Aright angels side ( side (rhs rhs) )ypotenuse (RHS) (RHS) angels hypotenuse and a hypotenuse and a ypotenuse and a side and a side

Similarity of triangles Similarity of triangles ??? ?????????? ? ????? ??? ?????????? ? ????? ?? ?????????? ? ????? ???? ???? ???? ???? ????

HERONS FORMULA HERON S FORMULA : the area A of a triangle with sides a,b andc units long and semi- perimeter (S) S= 1 2 ( a+b+c) is given by A = ? ? ? ? ?)(? ?) aA b c

If the angel between the sides a and b is ? , then the area A of a triangle is A = 1 2 ab sin ? Symmetrical properties of a circle Symmetrical properties of a circle o The line segment drawn from the center of a circle perpendicular to chord bisects the chord. o The line segment joining the center of a circle to the midpoint of a chord is perpendicular to the chord. PP AN = NB PQ 1v ST O O NM A B P Q QT READ THE ROOF PAGE 222 READ THE ROOF PAGE 222

If two tangent segments are drawn to a circle from an external point , then , The tangents are equal in length P O T Q The line segment joining the center to the external point bisects the angel between the tangents .

i.e 1. ?? = TQ and 2. M(<OTP) = M (<OTQ) Read the proof from your text page 244 Angle properties of a circle If ?? is a diameter of a circle , then they are ??? or ??? is called a semicircle . O A B An are is said to be a minor are , if it is less than a semicircle ; and a major are if it is greater than a semicircleX x . O 2x B C 2<BAC = <BOC same arc The measure of a central angel , angel subtended by an are is twice the measure of an inscribed angle in the circle subtended by the

Angeles inscribed in the same are of a circle [ i.e subtended by the same are ] are equal. o . The length ( the length l of an are that sub tenders an angel ? at the center of the circle with radius r is given by ? ? O A ??? 1800 ?? = ? ? = ? The area A of A sector with the central angel ? and radius r is given by A O AOB is area of sector A= ??2? 3600 ? ?