Sec_4.3-4.4_Apps-of-Prop_Thm

Sec_4.3-4.4_Apps-of-Prop_Thm - The Area Concept ,...

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The Area Concept , Similarity in Triangles, and Applications of the Basic Proportionality Theorem Section 4.3: The Area Concept (SMSG Postulates 17 – 20) The three properties which are necessary for all “area measure assignments”: 1) Every polygon is assigned a unique number called its “area”, 2) Congruent triangles (polygons) are assigned the same “area number”, 3) If a polygon P is the union of two regions R 1 and R 2 (which intersect only in a finite number of points and line segments), then Area of P = (Area of R 1 ) + (Area of R 2 ) Postulates 17, 18 and 19 ensure that the Euclidean Area Assignment meets requirements 1, 2 and 3, respectively. Postulate 20: The area of a rectangle is the product: Area = Base × Altitude. Theorem 4.3.1 (Proposition 35 in Book I of Euclid’s Elements ): Two parallelograms that share a common base and whose sides opposite this base are contained in the same line are equal in area. Proof: Consider parallelograms ABCD and EBCF that share the common base ± and whose opposite sides are contained in the same line ²³´² µ¶¶¶¶· . By the Converse of the Alternate Interior Angle Theorem, the angles indicated as congruent are congruent. Since the opposite sides of a parallelogram are congruent, the segments indicated as congruent are congruent. Now, AE = AD + DE = EF + DE = DE + EF = DF . Thus, ³¸ ¹´ and so, Δ ABE Δ DCF by ASA . Since two congruent figures have equal area measure, (Area of ΔABE) = (Area of ΔDCF) . Let Δ T 1 = Δ BCG and Δ T 2 = Δ DEG . Area of ABCD = (Area of ΔABE) + (Area of T 1 ) – (Area of T 2 ) and Area of EBCF = (Area of ΔDCF) + (Area of T 1 ) – (Area of T 2 ) . By substitution, Area of ABCD = (Area of ΔDCF) + (Area of T 1 ) – (Area of T 2 ) = Area of EBCF . QED T 2 T 1 G F D B A C E
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2 The Height of Triangles and Parallelograms (Relative to a chosen base) Definition: Given a triangle, choose one side to be called the base of the triangle. The height h of the triangle (relative to the chosen base) is the length of the segment drawn perpendicular to the base from the vertex not on the base. (See below.) Given a parallelogram, choose one side to be called the base of the parallelogram. The height h of the parallelogram (relative to the chosen base) is the length of the segment drawn perpendicular to the base from any point on side opposite the base. (See above.) The Altitude of a Triangle (Relative to one of the sides) Definition: The term “altitude of a triangle” has 3 meanings depending on the context.
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This note was uploaded on 11/02/2010 for the course MATH modern geo taught by Professor Shirley during the Spring '10 term at University of Texas.

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Sec_4.3-4.4_Apps-of-Prop_Thm - The Area Concept ,...

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