Question

The diameter of the circumcircle of a triangle with sides 5 cm, 6 cm and 7 cm is?
A) \[\dfrac{{3\sqrt 6 }}{2}cm\]
B) \[2\sqrt 6 cm\]
C) \[\dfrac{{35}}{{48}}cm\]
D) None of these

Answer 1

Hint:
Here we will use the basic concept of the circumcircle of the triangle. First, we will use the formula of the circumradius of the triangle and we will put the values of the sides of the triangle in it to get the circumradius. Then we will multiply it with 2 to get the diameter of the circumcircle of the triangle.

Formula used:
We will use the following formulas:
1) Circumradius \[R = \dfrac{{a \times b \times c}}{{4A}}\] where, \[A\] is the area of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle.
2) Area of triangle, \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \] , \[s\] is the semi perimeter of the triangle, and \[a\], \[b\] and \[c\] are the sides of the triangle.
3) Semi perimeter of the triangle\[s = \dfrac{{a + b + c}}{2}\], \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle.

Complete step by step solution:
It is given that there are three sides of the circle: 5 cm, 6 cm and 7 cm. Therefore, we get
\[a = 5\], \[b = 6\] and \[c = 7\]
First, we will find the value of the semi perimeter of the triangle.
Substituting \[a = 5\], \[b = 6\] and \[c = 7\] in the formula \[s = \dfrac{{a + b + c}}{2}\], we get
\[s = \dfrac{{5 + 6 + 7}}{2}\]
Adding the terms in the numerator, we get
\[ \Rightarrow s = \dfrac{{18}}{2}\]
Dividing 18 by 2, we get
\[ \Rightarrow s = 9\]
Now we will find the value of the area of the triangle.
Substituting \[s = 9\], \[a = 5\], \[b = 6\] and \[c = 7\] in the formula \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], we get
\[A = \sqrt {9\left( {9 - 5} \right)\left( {9 - 6} \right)\left( {9 - 7} \right)} \]
Subtracting the terms in the bracket, we get
\[ \Rightarrow A = \sqrt {9 \times 4 \times 3 \times 2} \]
Multiplying the terms, we get
\[ \Rightarrow A = \sqrt {216} = 6\sqrt 6 \]
Now we will find the circumradius of the triangle.
Substituting \[A = 6\sqrt 6 \], \[a = 5\], \[b = 6\] and \[c = 7\] in the formula \[R = \dfrac{{a \times b \times c}}{{4A}}\], we get
\[R = \dfrac{{5 \times 6 \times 7}}{{4\left( {6\sqrt 6 } \right)}}\]
Simplifying the terms, we get
\[ \Rightarrow R = \dfrac{{35}}{{4\sqrt 6 }}\]
We know that the radius is equal to the half of the diameter of the circle. Therefore, we will multiply circumradius by 2 to get the diameter of the circumcircle of the triangle, we get
Diameter of the circumcircle of the triangle \[ = 2R\]
Substituting \[R = \dfrac{{35}}{{4\sqrt 6 }}\] in the above equation, we get
\[ \Rightarrow \] Diameter of the circumcircle of the triangle \[ = 2 \times \dfrac{{35}}{{4\sqrt 6 }}\]
Multiplying the terms, we get
\[ \Rightarrow \] Diameter of the circumcircle of the triangle \[ = \dfrac{{35}}{{2\sqrt 6 }}\]
Now we will simplify this value by multiplying \[\sqrt 6 \] to the numerator and the denominator. Therefore, we get
\[ \Rightarrow \] Diameter of the circumcircle of the triangle \[ = \dfrac{{35}}{{2\sqrt 6 }} \times \dfrac{{\sqrt 6 }}{{\sqrt 6 }} = \dfrac{{35 \times \sqrt 6 }}{{2 \times 6}}\]
Multiplying the terms, we get
\[ \Rightarrow \] Diameter of the circumcircle of the triangle \[ = \dfrac{{35\sqrt 6 }}{{12}}\]
Hence, the value of the diameter of the circumcircle of the triangle is \[\dfrac{{35\sqrt 6 }}{{12}}\].

So, option D is the correct option.

Note:
Here we should note that the circumcircle is the circle whose circumference passes through all the vertices of the triangle or the polygon. Vertices are the point of the intersection of the two lines or the sides of the polygon. We have to note that the formula gives us the radius of the circumcircle so we have to double the value to get the value of the diameter of the circumcircle. As the formula of diameter is \[{\rm{diameter}} = 2 \times {\rm{radius}}\].