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8. Let T be an acute-angled triangle with vertices A, B and C. Let the sides BC, CA, AB of T have lengths a, b, c respectively with corresponding angles CÂB = a, ABC = ß, BĈA = 7. (a) Show that the area A of T is given by A = -be sin a. 1 2 (b) Now suppose further that the circumcircle S of T has diameter d and centre O. Show = d. (Hint: Consider the triangles AOB, BOC, COB where a b с that sin 3 sin y sin a AÔB = 2y etc.) abc (c) Deduce from parts (a) and (b) that A = 2d

8. Let T be an acute-angled triangle with vertices A, B and C. Let the sides BC, CA, AB of T have lengths a, b, c respectively with corresponding angles CÂB = a, ABC = ß, BĈA = 7. (a) Show that the area A of T is given by A = -be sin a. 1 2 (b) Now suppose further that the circumcircle S of T has diameter d and centre O. Show = d. (Hint: Consider the triangles AOB, BOC, COB where a b с that sin 3 sin y sin a AÔB = 2y etc.) abc (c) Deduce from parts (a) and (b) that A = 2d

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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8. Let T be an acute-angled triangle with vertices A, B and C. Let the sides BC, CA, AB of T have lengths a, b, c respectively with corresponding angles CÂB = a, ABC = ß, BĈA = 7. (a) Show that the area A of T is given by A - 2 (b) Now suppose further that the circumcircle S of T has diameter d and centre O. Show a b that sin B C sin y - = 1 sin a AÔB = 2y etc.) (c) Deduce from parts (a) and (b) that A (d) Now further suppose that a = 90°. 2Δ. bc be sin a. d. (Hint: Consider the triangles AOB, BOC, COB where abc 2d Show that a = d. Deduce that b² + c² = d² and √d² + 4A + √ 2 and c (e) Suppose that b≥ c. Show that b = (Hint: First calculate (b + c)² and (b − c)² using part (d) above). √d²-4A √d² +4A - √ď² – 4A 2
Transcribed Image Text:8. Let T be an acute-angled triangle with vertices A, B and C. Let the sides BC, CA, AB of T have lengths a, b, c respectively with corresponding angles CÂB = a, ABC = ß, BĈA = 7. (a) Show that the area A of T is given by A - 2 (b) Now suppose further that the circumcircle S of T has diameter d and centre O. Show a b that sin B C sin y - = 1 sin a AÔB = 2y etc.) (c) Deduce from parts (a) and (b) that A (d) Now further suppose that a = 90°. 2Δ. bc be sin a. d. (Hint: Consider the triangles AOB, BOC, COB where abc 2d Show that a = d. Deduce that b² + c² = d² and √d² + 4A + √ 2 and c (e) Suppose that b≥ c. Show that b = (Hint: First calculate (b + c)² and (b − c)² using part (d) above). √d²-4A √d² +4A - √ď² – 4A 2
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