Question

In \[\Delta ABC, AB = 6\sqrt 3 \] cm, \[AC = 12\] cm and \[BC = 6\] cm. The \[\angle B\] is:
A. \[{120^ \circ }\]
B. \[{60^ \circ }\]
C. \[{90^ \circ }\]
D. \[{45^ \circ }\]

Answer 1

Hint: In this question we have found that particular angle. Before finding the value of the required angle, we have to find the type of the given triangle.
By the length of the sides of the given triangle, we have to find whether it is a right-angle triangle or not.
To check the type of the triangle we will apply Pythagoras theorem.
Pythagoras theorem states that, for a right-angle triangle, the value of the square of the hypotenuse is the sum of the square of base and the square of perpendicular.
That means, \[{\text{Hypotenus}}{{\text{e}}^{\text{2}}}{\text{ = bas}}{{\text{e}}^{\text{2}}}{\text{ + perpendicula}}{{\text{r}}^{\text{2}}}\]

Complete step-by-step answer:
It is given that; in the \[\Delta ABC\], \[AB = 6\sqrt 3 {\text{ cm}}\], \[AC = 12\;{\text{cm}}\] and \[BC = 6{\text{ cm}}\].
We have to find the value of \[\angle B\].
First, we will check whether the triangle is a right-angle triangle.
To check the triangle, we will apply Pythagoras theorem.
Pythagoras theorem states that, for a right-angle triangle, the value of the square of the hypotenuse is the sum of the square of base and the square of perpendicular.
That means, \[{\text{Hypotenus}}{{\text{e}}^{\text{2}}}{\text{ = bas}}{{\text{e}}^{\text{2}}}{\text{ + perpendicula}}{{\text{r}}^{\text{2}}}\]
Again, if the above condition holds for any triangle, it means the triangle is a right-angle triangle.
Here, the longest side is \[AC = 12\]cm.
Now,
\[A{B^2} + B{C^2} = {(6\sqrt 3 )^2} + {6^2}\]
Simplifying we get,
\[A{B^2} + B{C^2} = 108 + 36 = 144\]
Again, \[A{C^2} = {12^2} = 144\]
So, it satisfies the Pythagoras theorem that,
\[A{B^2} + B{C^2} = A{C^2}\]
So, \[\Delta ABC\] is a right-angle triangle.
               

We know that, for a right-angle triangle, the opposite side of the hypotenuse is right angle.
So, the opposite side of \[AC\] is \[\angle B\].
So, \[\angle B = {90^ \circ }\]
Hence, the value of \[\angle B\] is \[{90^ \circ }\]

So, the correct answer is “Option C”.

Note: Pythagoras theorem states that, for a right-angle triangle, the value of the square of the hypotenuse is the sum of the square of base and the square of perpendicular.
That means, \[{\text{Hypotenus}}{{\text{e}}^{\text{2}}}{\text{ = bas}}{{\text{e}}^{\text{2}}}{\text{ + perpendicula}}{{\text{r}}^{\text{2}}}\]
Again, if the above condition holds for any triangle, it means the triangle is a right-angle triangle.