A triangle has sides of length 10, 17, and 21 units. What is the radius of its
circumscribed circle?
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2003.fall/Perry/Assign%204%20Triangle%20Proof/Triangle%20Proof.htm
Look at circumcenter.
Well, if this triangle was a person, I would say it likes to party! Just imagine it dancing around in a circumscribed circle. Now, to find the radius, we can use a little geometry humor.
Considering that the triangle is a right-angled triangle, we can say that half the length of the hypotenuse is equal to the radius of the circumscribed circle.
Now, let's find the hypotenuse of this right-angled triangle using the Pythagorean theorem: a² + b² = c². Here, a is the length of one side, which is 10, and b is the length of the other side, which is 17. So the hypotenuse, c, is the square root of (10² + 17²).
Grabbing my imaginary calculator... *beep boop beep* The hypotenuse, c, is approximately 19.21 units.
Since half the hypotenuse is equal to the radius of the circumscribed circle, we can say that the radius is approximately 9.60 units.
So, the radius of the circumscribed circle for this triangle is approximately 9.60 units. Now that's a hip and happening circumscribed circle!
To find the radius of the circumscribed circle of a triangle, we can use the formula:
radius = (abc) / (4A)
where a, b, and c are the side lengths of the triangle, and A is the area of the triangle.
First, let's find the area of the triangle using Heron's formula:
s = (a + b + c) / 2
where s is the semiperimeter of the triangle.
s = (10 + 17 + 21) / 2 = 48 / 2 = 24
Now we can calculate the area of the triangle using Heron's formula:
A = sqrt(s * (s - a) * (s - b) * (s - c))
A = sqrt(24 * (24 - 10) * (24 - 17) * (24 - 21))
A = sqrt(24 * 14 * 7 * 3)
A = sqrt(7056)
A ≈ 84.07
Now we can calculate the radius of the circumscribed circle:
radius = (abc) / (4A)
radius = (10 * 17 * 21) / (4 * 84.07)
radius = 3570 / 336.28
radius ≈ 10.63
Therefore, the radius of the circumscribed circle is approximately 10.63 units.
To find the radius of the circumscribed circle of a triangle, we can use the formula derived from the Law of Sines.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In other words:
a/sin(A) = b/sin(B) = c/sin(C)
Where:
a, b, c are the lengths of the sides of the triangle
A, B, C are the measures of the angles opposite sides a, b, c respectively
In this case, we have a triangle with sides of length 10, 17, and 21 units. Let's label them as follows:
a = 10
b = 17
c = 21
To find the radius of the circumscribed circle, we need to find the length of one side of the triangle. Let's find the length of side a using the Law of Sines.
We are given that the side opposite angle A has length 10, so:
a/sin(A) = 10/sin(A)
The next step is to find the measure of angle A. We can use the Law of Cosines to find this angle. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we know c = 21, a = 10, and b = 17. Let's solve for cos(C):
21^2 = 10^2 + 17^2 - 2(10)(17)*cos(C)
441 = 100 + 289 - 340*cos(C)
152 = -340*cos(C)
cos(C) = -152/340
cos(C) ≈ -0.4471
To find the measure of angle A, we can use the Law of Cosines again, this time solving for angle A:
a^2 = b^2 + c^2 - 2bc*cos(A)
10^2 = 17^2 + 21^2 - 2(17)(21)*cos(A)
100 = 289 + 441 - 714*cos(A)
-630 = -714*cos(A)
cos(A) ≈ -0.8821
Now that we have the measure of angle A (approximately cos(A) ≈ -0.8821), we can use the inverse cosine function (cos^-1) to find the angle itself:
A ≈ cos^-1(-0.8821)
A ≈ 150.12 degrees
Finally, we can substitute the values into the Law of Sines equation to solve for side a:
a/sin(A) = 10/sin(150.12)
a/0.5 = 10/(0.866)
a ≈ 10 * 0.5 / 0.866
a ≈ 5.77
Now that we have the length of side a, we can find the radius of the circumscribed circle using the formula:
R = (abc) / (4Δ)
Where:
R is the radius of the circumscribed circle
a, b, c are the lengths of the sides of the triangle
Δ is the area of the triangle
Using Heron's formula, we can find the area Δ of the triangle:
Δ = √(s(s-a)(s-b)(s-c))
Where:
s = (a+b+c)/2
In this case,
a = 5.77
b = 17
c = 21
Plugging in the values:
s = (5.77+17+21)/2
s ≈ 21.39
Now we can use the area formula:
Δ = √(21.39(21.39-5.77)(21.39-17)(21.39-21))
Δ ≈ √(21.39 * 15.62 * 4.39 * 0.39)
Δ ≈ √(501.34 * 2.73 * 0.39)
Δ ≈ √(358.042)
Δ ≈ 18.91
Substituting the values into the formula for the radius of the circumscribed circle:
R = (5.77 * 17 * 21) / (4 * 18.91)
R = 2157.27 / 75.64
R ≈ 28.53
Therefore, the radius of the circumscribed circle for the given triangle is approximately 28.53 units.
Damon's url is a nice proof and method of construction, but does not help much with calculating the circumradius.
R = abc/4A
Still not there, since the area A is unknown
There are various formulas for that, however, such as
Heron's formula, or
A = 1/2 ab sinC (find C using the law of cosines)