Pre-Calc Exam Notes 58

Pre-Calc Exam Notes 58 - K = 1 2 AC BD sin . 9. From...

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58 Chapter 2 General Triangles §2.4 Another formula 11 for the area of a triangle given its three sides is given below: For a triangle ABC with sides a b c , the area is: Area = K = 1 2 r a 2 c 2 p a 2 + c 2 b 2 2 P 2 (2.34) For the triangle in Example 2.16, the above formula gives an answer of exactly K = 1 on the same TI-83 Plus calculator that failed with Heron’s formula. Exercises For Exercises 1-6, ±nd the area of the triangle ABC . 1. A = 70 , b = 4, c = 12 2. a = 10, B = 95 , c = 35 3. A = 10 , B = 48 , C = 122 , c = 11 4. A = 171 , B = 1 , C = 8 , b = 2 5. a = 2, b = 3, c = 4 6. a = 5, b = 6, c = 5 7. Find the area of the quadrilateral in Figure 2.4.3 below. 2 4 3.5 6 5.5 Figure 2.4.3 Exercise 7 A B C D θ Figure 2.4.4 Exercise 8 8. Let ABCD be a quadrilateral which completely contains its two diagonals, as in Figure 2.4.4 above. Show that the area K of ABCD is equal to half the product of its diagonals and the sine of the angle they form, i.e.
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Unformatted text preview: K = 1 2 AC BD sin . 9. From formula (2.26) derive the following formula for the area of a triangle ABC : Area = K = a 2 sin B sin C 2 sin ( B + C ) 10. Show that the triangle area formula Area = K = 1 4 R ( a + ( b + c ))( c ( a b ))( c + ( a b ))( a + ( b c )) is equivalent to Herons formula. ( Hint: In Herons formula replace s by 1 2 ( a + b + c ) . ) 11. Show that the triangle area formula (2.34) is equivalent to Herons formula. ( Hint: Factor the expression inside the square root. ) 12. Find the angle A in Example 2.16, then use formula (2.23) to nd the area. Did it work? 11 Due to the Chinese mathematician Qiu Jiushao (ca. 1202-1261)....
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