The sides of a triangle are 8 cm, 10 cm, and 14 cm. Determine the radius of the circumscribed circle.
you can start here:
artofproblemsolving.com/wiki/index.php/Circumradius#Formula_for_Circumradius
To determine the radius of the circumscribed circle of a triangle, you can use the formula:
\[ R = \frac{abc}{4A} \]
where \( R \) is the radius of the circumscribed circle, \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle, and \( A \) is the area of the triangle.
To find the area of the triangle, we can use Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semiperimeter of the triangle given by:
\[ s = \frac{a + b + c}{2} \]
In this case, we have the sides of the triangle as 8 cm, 10 cm, and 14 cm. We can calculate the semiperimeter as:
\[ s = \frac{8 + 10 + 14}{2} = 16 \]
Next, we can use Heron's formula to find the area of the triangle:
\[ A = \sqrt{16(16-8)(16-10)(16-14)} = \sqrt{16(8)(6)(2)} = \sqrt{3072} \]
Now, we can substitute the values of \( a \), \( b \), \( c \), and \( A \) into the formula for the radius of the circumscribed circle:
\[ R = \frac{abc}{4A} = \frac{8 \times 10 \times 14}{4 \times \sqrt{3072}} \]
Evaluating this expression, we find:
\[ R \approx 5.742 \, \text{cm} \]
Therefore, the radius of the circumscribed circle of the given triangle is approximately 5.742 cm.