A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of a radius 3.what is the area of the triangle?
Area is 8.64squre units. Ratio of the side lengths 3:4:5 means it is a right angled triangle. Hypthenuse is the diameter of the circle, 6cm length. Other sides are 3.6cm and 4.8cm. So the area is 1/2 *3.6*4.8 = 8.64
To find the area of the triangle, we first need to determine its side lengths.
Since the side lengths are in the ratio 3:4:5, we can let the length of the shortest side be 3x, the length of the second side be 4x, and the longest side be 5x.
We know that the triangle is inscribed in a circle with a radius of 3. In an inscribed triangle, the diameter of the circle is equal to the length of the triangle's longest side.
So, in this case, the diameter of the circle is 2 times the radius, which is 6. Therefore, the longest side of the triangle, 5x, is equal to 6.
Setting up the equation:
5x = 6
Dividing both sides by 5, we get:
x = 6/5
Now, we can substitute this value of x back into our side lengths to find the lengths of the triangle:
Shortest side length = 3x = 3 * (6/5) = 18/5
Second side length = 4x = 4 * (6/5) = 24/5
Longest side length = 5x = 5 * (6/5) = 6
Now that we have the side lengths of the triangle, we can use Heron's formula to calculate its area.
Heron's formula states that the area of a triangle can be calculated using the lengths of its sides, using the formula:
Area = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, and a, b, and c are the side lengths.
The semi-perimeter (s) is calculated as:
s = (a + b + c) / 2
In our case, a = 18/5, b = 24/5, and c = 6. Plugging these values into the formula:
s = (18/5 + 24/5 + 6) / 2 = (48/5 + 6) / 2 = (54/5) / 2 = 27/5
Now, we can calculate the area of the triangle using Heron's formula:
Area = √(27/5 * (27/5 - 18/5) * (27/5 - 24/5) * (27/5 - 6))
Simplifying the expression inside the square root:
Area = √(27/5 * 9/5 * 3/5 * 21/5) = √(1323/625)
Taking the square root:
Area = √1323/√625
Simplifying:
Area = √(1764/625)
Area = √2.8224
Area ≈ 1.68
Therefore, the area of the triangle is approximately 1.68 square units.