If a circle is inscribed in a triangle with vertex angles 30o - 60o - 90o , then what is the area of the triangle divided by the area of the circle?
To find the area of the triangle divided by the area of the circle, we need to calculate the area of both the triangle and the circle.
Given that the triangle has vertex angles of 30°, 60°, and 90°, we can find the lengths of the triangle's sides using trigonometric ratios.
Let's assume that the triangle's side opposite the 30° angle is of length "a." Therefore, the side opposite the 60° angle is of length "a√3" (using the properties of a 30°-60°-90° triangle). And finally, the hypotenuse (side opposite the 90° angle) is of length "2a."
The radius of the inscribed circle in a triangle can be found by using the formula: r = (area of triangle) / (semiperimeter of triangle).
The semiperimeter of the triangle can be calculated as the sum of its three sides divided by 2.
Therefore, the semiperimeter, s = (a + a√3 + 2a) / 2 = (3 + √3) a / 2.
The area of the triangle, A = (base * height) / 2 = (a√3 * 2a) / 2 = √3 a^2.
Now, let's find the radius of the inscribed circle.
r = A / s = (√3 a^2) / ((3 + √3) a / 2) = (2√3 a^2) / (3 + √3) a = (2√3 a) / (3 + √3).
To find the area of the circle, we will use the formula for the area of a circle: A = πr^2.
Substituting the value of r, we get: A = π * [(2√3 a) / (3 + √3)]^2 = (12π a^2) / (9 + 6√3 + 3).
Finally, to find the area of the triangle divided by the area of the circle:
(Area of Triangle) / (Area of Circle) = (√3 a^2) / [(12π a^2) / (9 + 6√3 + 3)].
Simplifying further, we have:
(√3 a^2) / [(12π a^2) / (9 + 6√3 + 3)] = (√3 a^2) * [(9 + 6√3 + 3) / (12π a^2)].
Cancelling out the a^2 term, we get:
= (√3 / 12π) * [(9 + 6√3 + 3)].
Simplifying further, we have:
= (√3 / 12π) * (12 + 6√3).
= (√3 / π) * (2 + √3).
Therefore, the area of the triangle divided by the area of the circle is (√3 / π) * (2 + √3).
To find the area of the triangle divided by the area of the circle, we need to calculate both areas separately.
Step 1: Find the area of the triangle.
Since the triangle has a vertex angle of 30°-60°-90°, we know that the shortest side is opposite the 30° angle and the hypotenuse is opposite the 90° angle.
Step 2: Calculate the area of the triangle.
The area of a triangle can be found using the formula A = (1/2) * base * height. In this case, the base is the shortest side of the triangle (opposite the 30° angle) and the height is the side opposite the 60° angle.
Let's assume the shortest side of the triangle is 'a'. The height of the triangle can be found using the trigonometric relationship: height = a * sin(60°).
Substituting the values, we have A_triangle = (1/2) * a * a * sin(60°) = (1/2) * a^2 * √3 / 2 = a^2 * √3 / 4.
Step 3: Find the area of the circle.
To find the area of the circle, we need to know the radius. The radius of the inscribed circle can be found by drawing perpendiculars from the center of the circle to the sides of the triangle. These perpendiculars are equal in length and bisect the sides of the triangle.
Since the hypotenuse (opposite the 90° angle) is twice the length of the shortest side, the radius of the inscribed circle is half of the shortest side, i.e., r = a/2.
Using the formula for the area of a circle, A_circle = π * r^2, we can substitute the value of the radius into the formula to find the area of the circle.
A_circle = π * (a/2)^2 = π * a^2/4.
Step 4: Divide the area of the triangle by the area of the circle.
Now, we can calculate the ratio of the areas by dividing the area of the triangle by the area of the circle:
A_triangle/A_circle = (a^2 * √3 / 4) / (π * a^2/4) = √3 / π.
Therefore, the area of the triangle divided by the area of the inscribed circle is √3 / π.