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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?
Wiki User
∙ 14y ago
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, and
BD = (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
Wiki User
∙ 14y ago
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The center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle?
true
How is the Pythagorean theorem used to develop the equation of a circle in standard form?
The Pythagorean theorem is used to develop the equation of the circle. This is because a triangle can be drawn with the radius and any other adjacent line in the circle.
How do you find the value of x on a polygon?
It depends very much on what x is. Whether it is a side or a diagonal or a radius of a circumscribing circle or circumscribed incircle, an apothem, an interior or exterior angle or some other measure. It also depends on the number of sides that the polygon has.
What is the formula for the area of a regular hexagon?
A regular hexagon can be considered as being built up of six equilateral triangles. Each equilateral triangle has an area of (b/2) * sqrt (3b/2) where b is the side of the equilateral triangles that make up the hexagon and also the radius of the hexagon's circumscribed circle, and sqrt means the square root ofSo the area of the regular hexagon with side length b is 3 * b * sqrt (3b/2)
Is a circle graph a function?
No, a circle graph is never a function.
To circumscribed a circle about a triangle you use the?
To circumscribed a circle about a triangle you use the angle. This is to get the right measurements.
How many circumscribed circles can a triangle have?
A triangle has exactly one circumscribed circle.
How do you find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r?
It is 5.196*r^2 square units.
Is the incenter of a triangle the center of the circle circumscribed about the triangle?
No.
If a circle is circumscribed about a triangle the center of the circle is called the of the triangle?
Centre
Is the circumcenter of a triangle the center of the circle circumscribed about the triangle?
Yes.
The of a triangle is the center of the only circle that can be circumscribed about it.?
no
The circumcenter of a triangle is the center of the only circle that can be it?
circumscribed about
The center of the circumscribed circle about a triangle is equidistant to the vertices of the inscribed triangle?
true
How would you construct an isosceles triangle if only given the vertex angle and the radius of the circumscribed circle?
You have an isosceles triangle, and a circle that is drawn around it. You know the vertex angle of the isosceles triangle, and you know the radius of the circle. If you use a compass and draw the circle according to its radius, you can begin your construction. First, draw a bisecting cord vertically down the middle. This bisects the circle, and it will also bisect your isosceles triangle. At the top of this cord will be the vertex of your isosceles triangle. Now is the time to work with the angle of the vertex. Take the given angle and divide it in two. Then take that resulting angle and, using your protractor, mark the angle from the point at the top of the cord you drew. Then draw in a line segment from the "vertex point" and extend it until it intersects the circle. This new cord represents one side of the isosceles triangle you wished to construct. Repeat the process on the other side of the vertical line you bisected the circle with. Lastly, draw in a line segment between the points where the two sides of your triangle intersect the circle, and that will be the base of your isosceles triangle.
The circumcenter of a triangle is the center of the only circle that can be circumscribed about it?
True
Is the circumcenter of a triangle the center of the only circle that can be circumscribed?
Yes, it is.
