Question

The circumradius of the triangle whose sides are \[13, 12\] and \[5\], is
A) 15
B) \[\dfrac{{13}}{2}\]
C) \[\dfrac{{15}}{2}\]
D) 6

Answer 1

Hint:
Here, we have to find the circumradius of the triangle. Circumradius of a triangle is the radius of the circumcircle of a triangle, or the line segment connecting any vertex of a triangle to the circumcenter of the triangle. The Circumcircle of a triangle is a circle that passes through all vertices of a triangle.
Formula used: Length of a Circumradius of a triangle \[R = \dfrac{{abc}}{{\sqrt {(a + b + c)(b + c - a)(c + a - b)(a + b - c)} }}\] where \[a,b,c\] are the sides of a triangle.

Complete step by step solution:
We are given that the sides of a triangle are 13,12 and 5cm.
Now, by using the formula, \[R = \dfrac{{abc}}{{\sqrt {(a + b + c)(b + c - a)(c + a - b)(a + b - c)} }}\], we have
\[ \Rightarrow R = \dfrac{{13 \times 12 \times 5}}{{\sqrt {(13 + 12 + 5)(12 + 5 - 13)(5 + 13 - 12)(13 + 12 - 5)} }}\]
\[ \Rightarrow R = \dfrac{{780}}{{\sqrt {(30)(4)(6)(20)} }}\]
Multiplying the terms, we get
\[ \Rightarrow R = \dfrac{{780}}{{\sqrt {600 \times 4 \times 6} }}\]
Rewriting the terms, we get
\[ \Rightarrow R = \dfrac{{780}}{{\sqrt {6 \times 100 \times 6 \times 4} }}\]
Simplifying the terms, we get
\[ \Rightarrow R = \dfrac{{780}}{{6 \times 10 \times 2}}\]
Multiplying the terms in the denominator, we get
\[ \Rightarrow R = \dfrac{{780}}{{120}}\]
Dividing the terms, we get
 \[ \Rightarrow R = 6\]

Therefore, the circumradius of a triangle whose sides are 13, 12 and 5 cm, is 6.

Note:
The circumradius of a cyclic polygon is the radius of the circumscribed circle of that polygon. For a triangle, it is the measure of the radius of the circle that circumscribes the triangle. Since every triangle is cyclic, every triangle has a circumscribed circle, or a circumcircle.