Construct an Equilateral Triangle
How to construct an Equilateral Triangle?
42. Construct an Equilateral Triangle whose area is equal to the area of a scalene triangle ABC. (Page 433 of Geometry).
There are three steps to solve this problem. Suppose the scalene triangle is characterized with a base b and a height h. The first step involves a semicircle of the diameter (b+h) with an inscribed right triangle. The last two steps involve the geometric means of right triangles.
(1) As we know from the Corollary 2 of Theorem 9-7 that an angle inscribed in a semicircle is a right angle. The two sides of the right angle and the diameter form a right triangle. Let the diameter of the semicircle equals to the length of b+h of the given scalene triangle.
(2) We know from the corollary 1 of Theorem 8-1 that if the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric means between the two segments of the hypotenuse. That is to say x = sqrt (b*h).
(3) We know from the corollary 2 of Theorem 8-1 that if the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric means between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. That is to say x = sqrt (k*s), where k and s are the altitude and base of the equilateral triangle we need to construct. This right triangle should be a special 30-60-90 right triangle. The x is the length of one leg of the special right triangle.
Since we know the length of each side of the equilateral triangle, it should be easy to construct the equilateral triangle.
CHRISRAO/饶旭东
42. Construct an Equilateral Triangle whose area is equal to the area of a scalene triangle ABC. (Page 433 of Geometry).
There are three steps to solve this problem. Suppose the scalene triangle is characterized with a base b and a height h. The first step involves a semicircle of the diameter (b+h) with an inscribed right triangle. The last two steps involve the geometric means of right triangles.
(1) As we know from the Corollary 2 of Theorem 9-7 that an angle inscribed in a semicircle is a right angle. The two sides of the right angle and the diameter form a right triangle. Let the diameter of the semicircle equals to the length of b+h of the given scalene triangle.
(2) We know from the corollary 1 of Theorem 8-1 that if the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric means between the two segments of the hypotenuse. That is to say x = sqrt (b*h).
(3) We know from the corollary 2 of Theorem 8-1 that if the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric means between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. That is to say x = sqrt (k*s), where k and s are the altitude and base of the equilateral triangle we need to construct. This right triangle should be a special 30-60-90 right triangle. The x is the length of one leg of the special right triangle.
Since we know the length of each side of the equilateral triangle, it should be easy to construct the equilateral triangle.
CHRISRAO/饶旭东
Labels: triangle
posted by Unknown at 7:23 PM
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