Question

The statement that circumradius and radius of a triangle are 12 and 8 respectively cannot be correct.
Reason
Circum radius \[ \ge \] in radius
A) Both the statements are TRUE and reason is the correct expansion of assertion
B) Both the statements are TRUE but reason is NOT the correct explanation of assertion
C) assertion is TRUE and reason is FALSE
D) assertion is FALSE and reason is TRUE

Answer 1

Hint:
Here we need to check whether the given statement is correct or not. We will use the relation between the radius and the circum radius of the triangle to check whether the given circum radius is greater than or equal to two times the in radius or not. If it satisfies the condition then the given statement is correct otherwise incorrect.

Complete Step by Step Solution:
We know that the circumcenter of the triangle is greater than or equal to the in center of the triangle i.e.
Circum radius \[ \ge \] in radius ……………………………\[\left( 1 \right)\]
The given circum radius of the triangle is 12 and the in radius of the triangle is 8.
Now, we will substitute the value of radius in equation \[\left( 1 \right)\]. Therefore, we get
\[ \Rightarrow \] Circum radius \[\ge 2\times 8\]
On multiplying the numbers, we get
\[ \Rightarrow \] Circum radius \[\ge 16\]
So, the circumcenter should be greater than or equal to 16 but the given circum center is less than 16.
So, the given assertion the radius and in radius of a triangle are 12 and 8 respectively cannot be correct is true.
The given statement is true but the reason is not the correct explanation of the given assertion.

Hence, the correct option is option B.

Note:
To solve this question, we need to know the meaning of the terms circum center and the in radius of the triangle. The in radius of a triangle is formed by lines which are angle bisectors of all three angles. The point at which these three lines meet is called as the center of the in circle, and the in radius is a line drawn from the center to perpendicularly intersect a side of the triangle. Circumcenter is defined as the center of the circle which surrounds the triangle or circumscribes the triangle.