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I assume that the "diameter of a triangle" means the diameter of the smallest circle containing it.
Assume the triangle has sides a, b and c and angles A, B and C (where angle A is opposite side a etc.). Calculate angle A from the law of cosines. Then:
a^2 = b^2 +c^2 - 2*b*c*cos(A) so:
A = arccos((b^2 + c^2 - a^2)/(2*b*c)) -------------------(1)
Then, using the usual construction for finding the center, O, of the circumscribed circle (i.e. a perpendicular bisectors of each side is constructed and all three meet at point O) we can draw the radius from O to A. This will divide angle A into two smaller angles (in some cases one angle will be larger than A and the other will be negative, but they will still add up to angle A). Using simple trigonometric formulas we can calculate angle A by adding the two together. The result then is:
A = arccos(b/(2*r)) + arccos(c/(2*r)) -----------------------------(2)
If we then equate the right hand sides of (1) and (2) we get an equation involving a, b, c and r. These can be solved to find r in terms of a,b and c. I used the easy way using Maple and got the result shown below.
r = abs[sqr(-b^4+2*b^2*c^2+2*b^2*a^2-c^4+2*c^2*a^2-a^4)*a*c*b]/
[ b^4-2*b^2*c^2-2*b^2*a^2+c^4-2*c^2*a^2+a^4]
The diameter is, of course, 2*r.
This answer checks the cases for an equilateral triangle and for a 3,4,5 right triangle, each of which is easy to do by much simpler methods.
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∙ 12y ago
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What is a triangle in a circle that is formed by two chords and a diameter?
If a triangle is drawn in a circle with a diameter as the base of the triangle, then the angle opposite that diameter is a right angle. This is an extension of the theorem that the angle which an arc of a circle subtends at the centre of a circle is twice the angle which the arc subtends at the circumference. In the case of a diameter, then the angle subtended at the centre is 180° and thus the angle at the circumference is 90°.
What is the area of a Reuleaux triangle that has a diameter of 4.5 in?
It is 1.2336 square inches, approx.
Can a triangle be obtained on the cross section of a cone?
Yes if cut in half from its apex to its base diameter.
What relationship does the hypotenuse have with the circle?
The hypotenuse has no intrinsic relationship to the circle. The hypotenuse is the side of a right triangle that is opposite to the right angle. You can draw a circle that has a hypotenuse as its diameter or its radius, but you can do that with any line segment. It would not be related in another way to the triangle.
Does a diameter that is perpendicular to a chord bisect the chord?
Yes, any diameter which is perpendicular to a chord bisects said chord. This can be proved most easily with a picture, but is proved using a congruent triangle proof. Both triangles include the points at the center of the circle and the intersection of the diameter and chord. The other points should be the endpoints of the chord. They are congruent by hypotenuse leg; it was given that they are right triangle by the "perpendicular", the "leg" is the segment between the center of the circle and the intersection, and it is equal in both triangles because it is the same segment in both triangles. The hypotenuses are equal because both are radii of the circle. Because the triangles are congruent, their sides must be so the two halves of the chord are congruent, and therefore the chord is bisected by the diameter.
Does a triangle have a diameter?
No, a triangle is defined by its three sides and/or its angels. The diameter is a characteristic of a circle.
What is the diameter of a circle with a triangle in it?
It could be anything. Just because it has a triangle in it doesn't in any way determine its diameter.
What is the perimeter of a triangle which a diameter of 8 inches?
Triangles don't have a diameter. They have a base and a height.
It a right triangle is inscribed in a circle the the what is a diameter of the circle?
The diameter is the distance across the centre of the circle.
When a circle is drawn through each vertex of a right triangle the triangle's hypotenuse will be equal to?
Its diameter.
What is a triangle in a circle that is formed by two chords and a diameter?
If a triangle is drawn in a circle with a diameter as the base of the triangle, then the angle opposite that diameter is a right angle. This is an extension of the theorem that the angle which an arc of a circle subtends at the centre of a circle is twice the angle which the arc subtends at the circumference. In the case of a diameter, then the angle subtended at the centre is 180° and thus the angle at the circumference is 90°.
What size eqiulateral triangle fits into a circle with a diameter of 5.5 inches?
A triangle with sides of 4.76 inches or less.
When a right triangle is inscribed in a circle will one of the legs of the triangle always be the diameter of a circle?
yes. the leg of the triangle has to be formed different because of the circle
When a circle is drawn through each vertex of a right triangle the triangles hypotenuse will be equal to what?
The length of the circle's diameter
What is the area of a Reuleaux triangle that has a diameter of 4.5 in?
It is 1.2336 square inches, approx.
How do you draw an isoscles right angled triangle?
An isosceles triangle is usually drawn with the two sides of equal length as the legs and the third side as the base. For a right angled isosceles triangle then the hypotenuse is drawn as the base with the two sides of equal length as the legs joining together at a right angle. Draw a circle. Draw a horizontal diameter with a second diameter perpendicular to the first. The hypotenuse is the horizontal diameter. Draw lines from the ends of this diameter to the point where one end of the second diameter meets the circumference. These are the two equal legs of the isosceles triangle. These legs meet at an angle of 90° .
What is true about the diameter of a circle?
It is 1/pi times the circumference. A triangle with the diameter as its hypotenuse and the third point anywhere on the circle is always a right-angled triangle. A quadrilateral with all four corners on a circle is a cyclic quadrilateral. If one of its diagonals is a diameter of the circle, it has two right angles.
