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The center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle?
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What are the dimensions of an isosceles triangle of least area that can be circumscribed about a circle of radius r?
The isosceles triangle of least area that can be circumscribed about a circle of radius r turns out to be not just isosceles, but also equilateral. Each side has length 2r x ( 3 )0.5 . The area is r2 x (27)0.5 . Thanks are due to litotes for pointing out that the original answer did not actually answer the question ! tpm Since the equilateral triangle is also an isosceles triangle, we can say that at least area that can be circumscribed to a circle is the area of an equilateral triangle.If we are talking only for isosceles triangle where base has different length than two congruent sides, we can say that at least area circumscribed to a circle with radius r, is the area of an isosceles triangle whose base angles are very close to 60 degrees. Solution: Let say that the isosceles triangle ABC is circumscribed to a circle with radius r, where BA = BC. We know that the center of the circle inscribed to a triangle is the point of the intersection of the three angle bisectors of the triangle. Let draw these angle bisectors, and denote with D the point where the bisector drawn from the vertex, B, of the triangle, intersects the base AC. Since the triangle is an isosceles triangle, then BD bisects the base and it is perpendicular to the base. So that AD = DC, OD = r, and the triangles ADB and AOD are right triangles (O is the center of the circle). In the triangle ADB, we have:tan A = BD/AD, so that AD = BD/tan A In the triangle AOD, we have:tan A/2 = OD/AD, so that AD = r/tan A/2, and AC = 2r/tan A/2 Therefore,BD/tan A = r/tan A/2, andBD = (r tan A)/tan A/2 Area of triangle ABC = (1/2)(AC)(BD) = (1/2)(2r/ tan A/2)[(r tan A)/tan A/2] = (r2 tan A)/tan2 A/2 After we try different acute angles measure, we see that the smallest area would be: If the angle A= 60⁰,then the Area of the triangle ABC = r2 tan 60⁰/tan2 30⁰ ≈ 5.1961r2 If the angle A= 59.8⁰,then the Area of the triangle ABC = (r2 tan 59.8⁰)/tan2 29.9⁰ ≈ 5.1962r2
What is the area of a triangle with base 12cm and height?
The area of a triangle can be calculated using the formula: Area = 0.5 x base x height. Given the base is 12cm and the height is unknown, the area cannot be determined without the height measurement. To find the area, you would need to know the specific value of the height in order to plug it into the formula and calculate the area of the triangle accurately.
What is sin y?
The Sine function (abbreviated sin) takes an angle and gives a ratio which is based on the sides of a right triangle. If you have a right triangle, and one of the angles (not the right angle) is labeled y then sin y equals the length of the side opposite of angle y divided by the length of the hypotenuse. The hypotenuse of a right triangle is the longest side, and is always opposite of the right angle.
How do you calculate the area of a sphere?
Surface Area (Sphere_ = 4 X pi X r^(2) = 4pi r(2) Volume (sphere) = (4/3)pi r^)3)
