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8. (a) An isosceles triangle T has two sides of length d and the included angle between these sides has size 0 (radians), say. Determine the area of T and the value of 0 for which T has the maximal area for a given value of d. (b) Let C be a circle of radius 1 and centre O. Let X, Y and Z be three distinct points on C. Determine the maximal possible area of the triangle XYZ assuming that it is right-angled. (c) Now let A, B and C be three distinct points on the circle C of radius 1 and suppose that the sides AB, AC and BC of the triangle ABC have lengths x, x and y respectively. Thus the angles ABC = AĈB = ß say. Let D be the point on C such that BD is a diameter of C. Show that the angle ADB : %3D and hence deduce that x = 2 sin B.

8. (a) An isosceles triangle T has two sides of length d and the included angle between these sides has size 0 (radians), say. Determine the area of T and the value of 0 for which T has the maximal area for a given value of d. (b) Let C be a circle of radius 1 and centre O. Let X, Y and Z be three distinct points on C. Determine the maximal possible area of the triangle XYZ assuming that it is right-angled. (c) Now let A, B and C be three distinct points on the circle C of radius 1 and suppose that the sides AB, AC and BC of the triangle ABC have lengths x, x and y respectively. Thus the angles ABC = AĈB = ß say. Let D be the point on C such that BD is a diameter of C. Show that the angle ADB : %3D and hence deduce that x = 2 sin B.

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.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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8. (a) An isosceles triangle T has two sides of length d and the included angle between these sides has size 0 (radians), say. Determine the area of T and the value of 0 for which T has the maximal area for a given value of d. (b) Let C be a circle of radius 1 and centre O. Let X, Y and Z be three distinct points on C. Determine the maximal possible area of the triangle XY Z assuming that it is right-angled. (c) Now let A, B and C be three distinct points on the circle C of radius 1 and suppose that the sides AB, AC and BC of the triangle ABC have lengths x, x and y respectively. Thus the angles ABC = AĈB = B say. Let D be the point on C such that BD is a diameter of C. Show that the angle ADB = B and hence deduce that x = 2 sin B. (d) By using the sine rule, or otherwise, show that y/x y2 /a? + 2? = 4. = 2 cos B and hence deduce that
Transcribed Image Text:8. (a) An isosceles triangle T has two sides of length d and the included angle between these sides has size 0 (radians), say. Determine the area of T and the value of 0 for which T has the maximal area for a given value of d. (b) Let C be a circle of radius 1 and centre O. Let X, Y and Z be three distinct points on C. Determine the maximal possible area of the triangle XY Z assuming that it is right-angled. (c) Now let A, B and C be three distinct points on the circle C of radius 1 and suppose that the sides AB, AC and BC of the triangle ABC have lengths x, x and y respectively. Thus the angles ABC = AĈB = B say. Let D be the point on C such that BD is a diameter of C. Show that the angle ADB = B and hence deduce that x = 2 sin B. (d) By using the sine rule, or otherwise, show that y/x y2 /a? + 2? = 4. = 2 cos B and hence deduce that
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