tag:blogger.com,1999:blog-38193052.post3311480907154583074..comments2024-03-20T08:57:17.447-03:00Comments on Jornalheiros: Geometria – Três triângulosPC Filhohttp://www.blogger.com/profile/16547063456626761789noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-38193052.post-27238992418819977762020-06-28T16:16:34.034-03:002020-06-28T16:16:34.034-03:00Since one side of the equilateral triangle is the ...Since one side of the equilateral triangle is the hypotenuse of the right triangle whose leg lengths are given, we can calculate its length:<br /><br />side of equilateral triangle = √(2² + 3²) = √13<br /><br />(By the way, this is also the length of the other right triangle’s hypotenuse, since that segment is another side of the equilateral triangle.)<br /><br />Now we calculate the sine and cosine of the larger acute angle of the right triangle on the right side of the figure (let’s call this angle measure θ):<br /><br />sin θ = 3/√13, cos θ = 2/√13<br /><br />Since the larger acute angles of the right triangles and one of the equilateral triangle’s angles make up a straight angle, that means that they sum to 180°. So, the right triangles’ larger acute angles sum to 120°, since each angle of an equilateral triangle measures 60° and 180° – 60° = 120°. We can then calculate the sine and cosine of the larger acute angle of the right triangle on the left:<br /><br />sin (120° – θ) = sin 120° cos θ – cos 120° sin θ<br />= (√3/2)(2/√13) – (–1/2)(3/√13)<br />= √(3/13) + (3/2)/√13<br />= (2√3 + 3)/(2√13)<br />cos (120° – θ) = cos 120° cos θ + sin 120° sin θ<br />= (–1/2)(2/√13) + (√3/2)(3/√13)<br />= –1/√13 + (3√3)/(2√13)<br />= (3√3 – 2)/(2√13)<br /><br />So, the legs of the right triangle on the left have lengths √3 + 3/2 and (3/2)√3 – 1.<br /><br />We now compute the areas of the triangles:<br />Area of triangle on the right = (1/2)(3)(2) = 3 square units<br />Area of triangle on the left = (1/2)[(3/2)√3 – 1](√3 + 3/2) = (5/8)√3 = 1.083 square units<br />Area of equilateral triangle = (√3/4)(√13)² = (13/4)√3 = 5.629 square units<br /><br />Since 5.629 > 3 + 1.083, the equilateral triangle has more area than the combined area of the right triangles.<br /><br />My answer: The equilateral trianglejrh150482https://www.blogger.com/profile/10502831081969372299noreply@blogger.com