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NOT TO SCALE a y S В The diagram shows a triangular piece of land ABC with dimensions AB =c metres, AC=b metres and BC=a metres, where a

NOT TO SCALE a y S В The diagram shows a triangular piece of land ABC with dimensions AB =c metres, AC=b metres and BC=a metres, where a

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter5: Similar Triangles
Section5.4: The Pythagorean Theorem
Problem 13E
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NOT TO SCALE b. a Z. y B The diagram shows a triangular piece of land ABC with dimensions AB =c metres, AC=b metres and BC=a metres, where a <b<c. The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let S and T be points on AB and AC respectively so that ST divides the land into two pieces of equal area. Let AS=x metres, AT=y metres and ST=z metres. (i) Show that xy=- bc. 2 (ii) Use the cosine rule in triangle AST to show that b?c? z? = x? + - bc cos A. 4x2 (iii) Show that the value of z² in the equation in part (ii) is a minimum when bc X = 2 (Р-2b)(Р—2с) (iv) Show that the minimum length of the fence is metres, 2 where P=a+b+c. (You may assume that the value of x given in part (iii) is feasible.)
Transcribed Image Text:NOT TO SCALE b. a Z. y B The diagram shows a triangular piece of land ABC with dimensions AB =c metres, AC=b metres and BC=a metres, where a <b<c. The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let S and T be points on AB and AC respectively so that ST divides the land into two pieces of equal area. Let AS=x metres, AT=y metres and ST=z metres. (i) Show that xy=- bc. 2 (ii) Use the cosine rule in triangle AST to show that b?c? z? = x? + - bc cos A. 4x2 (iii) Show that the value of z² in the equation in part (ii) is a minimum when bc X = 2 (Р-2b)(Р—2с) (iv) Show that the minimum length of the fence is metres, 2 where P=a+b+c. (You may assume that the value of x given in part (iii) is feasible.)
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