A triangle has corners at #(1 ,1 )#, #(7 ,9 )#, and #(4 ,2 )#. What is the area of the triangle's circumscribed circle?
Area of triangle's circumscribed circle is
By signing up, you agree to our Terms of Service and Privacy Policy
To find the area of the circumscribed circle of a triangle, we first need to determine the radius of the circumscribed circle. The radius ( R ) of the circumscribed circle is given by the formula:
[ R = \frac{abc}{4A} ]
where ( a, b, c ) are the lengths of the sides of the triangle, and ( A ) is the area of the triangle.
First, we calculate the lengths of the sides of the triangle using the given coordinates:
- Side ( a ) is between (1, 1) and (7, 9).
- Side ( b ) is between (7, 9) and (4, 2).
- Side ( c ) is between (4, 2) and (1, 1).
Using the distance formula ( d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} ), we find:
- Side ( a ): ( d_a = \sqrt{(7 - 1)^2 + (9 - 1)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 )
- Side ( b ): ( d_b = \sqrt{(4 - 7)^2 + (2 - 9)^2} = \sqrt{9 + 49} = \sqrt{58} )
- Side ( c ): ( d_c = \sqrt{(4 - 1)^2 + (2 - 1)^2} = \sqrt{9 + 1} = \sqrt{10} )
Next, we calculate the area ( A ) of the triangle using Heron's formula:
[ A = \sqrt{s(s - a)(s - b)(s - c)} ]
where ( s ) is the semiperimeter given by ( s = \frac{a + b + c}{2} ).
Plugging in the values:
[ s = \frac{10 + \sqrt{58} + \sqrt{10}}{2} ]
[ A = \sqrt{\frac{10 + \sqrt{58} + \sqrt{10}}{2} \cdot \left(\frac{10 + \sqrt{58} + \sqrt{10}}{2} - 10\right) \cdot \left(\frac{10 + \sqrt{58} + \sqrt{10}}{2} - \sqrt{58}\right) \cdot \left(\frac{10 + \sqrt{58} + \sqrt{10}}{2} - \sqrt{10}\right)} ]
After calculating ( A ), 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} ]
Finally, once we have the radius ( R ), we can find the area of the circumscribed circle using the formula ( \text{Area} = \pi R^2 ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- How do you find the equation of a circle in standard form given C(-2,8) and r=4?
- A triangle has corners at #(5 , 2 )#, #(1 ,3 )#, and #(7 ,4 )#. What is the radius of the triangle's inscribed circle?
- A circle's center is at #(9 ,3 )# and it passes through #(2 ,1 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?
- A triangle has corners at #(2 , 5 )#, #(4 ,8 )#, and #(4 ,6 )#. What is the radius of the triangle's inscribed circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/3 #, and the triangle's area is #25 #. What is the area of the triangle's incircle?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7