- #1
lizzie
- 25
- 0
a chord AB of a circle subtends an angle that is not equal to 60 degees at a point C on the circumference. ABC has maximum area. then find A & B in terms of the angle.
chaoseverlasting said:Can you try to find the area of the triangle in terms of the radius of the circle and the angle subtended? If you can do so, then its a simple problem in maximization.
For example, say the radius of the circle is r and the angle subtended at the point C is [tex]\theta[/tex] and the center of the circle is at the point [tex]O[/tex]. Also, the area of the triangle depends on the base and the height of the triangle, for the height to be maximum, the triangle MUST be isosceles for a given base, can you see why?
lizzie said:I feel in an isosceles triangle the base will be less but height will be more so how do we know that the area is maximum.
To find the area of a triangle in a circle, you can use the formula A = (1/2) * b * h, where b is the length of the base of the triangle and h is the height of the triangle. The base of the triangle should be a chord of the circle, and the height can be found by drawing a perpendicular line from the center of the circle to the base of the triangle.
No, the triangle must be inscribed in the circle, meaning that all three vertices of the triangle must lie on the circumference of the circle. This is necessary for the formula A = (1/2) * b * h to be valid.
The length of the base of the triangle can be found by using the chord theorem, which states that the length of a chord is equal to the diameter of the circle multiplied by the sine of half of the central angle that the chord subtends. In other words, b = 2r * sin(θ/2), where r is the radius of the circle and θ is the central angle.
If the triangle is not exactly inscribed in the circle, you can still use the formula A = (1/2) * b * h as long as the base of the triangle is a chord of the circle. However, the resulting area may not be completely accurate. To improve accuracy, you can divide the triangle into smaller triangles that are inscribed in the circle and find the area of each smaller triangle separately.
No, this formula only applies to triangles inscribed in a circle. To find the area of other shapes inscribed in a circle, different formulas and methods may need to be used.