SECTION 8.4 Area of a Triangle 577 ‘Are You Prepared?’ Answers 1. d x x y y 2 1 2 2 1 2 ( ) ( ) = − + − 2. 45 or 4 θ π = ° 74. Find f x 1( ) − if f x A x A 5 2 , 0. ( ) = + ≠ 75. If F x x x C 3 3 3 ( ) = − + + and a b , 1, 2 , [ ] [ ] = find F b F a . ( ) ( ) − 76. Simplify: x x x 4 3 ln 3 4 3 1 2 x x 1 2 1 2 2 ( ) ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ − 77. Solve: x x 4 3 1 − ≥ + 78. Convert 96° to radians. 79. If f x ax x2 5 2 ( ) = − + and a 0, < in which quadrant is the vertex located? How many x -intercepts does the graph of f have? In this section, several formulas for calculating the area of a triangle are derived. 8.4 Area of a Triangle Now Work the ‘Are You Prepared?’ problems on page 580. • Geometry Formulas (Section A.2, pp. A15–A16) • Using Half-angle Formulas to Find Exact Values (Section 7. 6, pp. 528–530) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find the Area of SAS Triangles (p. 577) 2 Find the Area of SSS Triangles (p. 578) THEOREM Area of a Triangle The area K of a triangle is = K bh 1 2 (1) where b is the base and h is the altitude (perpendicular to b ) drawn to that base. Proof Look at the triangle in Figure 39.Around the triangle construct a rectangle of altitude h and base b, as shown in Figure 40. Triangles 1 and 2 in Figure 40 are equal in area, as are triangles 3 and 4. Consequently, the area of the triangle with base b and altitude h is exactly half the area of the rectangle, which is bh. Figure 40 h b 2 3 1 4 Figure 39 h b ■ 1 Find the Area of SAS Triangles If the base b and the altitude h to that base are known, then the area of the triangle can be found using formula (1). Usually, though, the information required to use formula (1) is not given. Suppose, for example, that two sides a and b and NOTE Typically, A is used for area. However, because A is also used as the measure of an angle, K is used here for area to avoid confusion. j
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