Question

Tick the correct answer and justify: In $\Delta ABC,AB = 6\sqrt 3 cm,AC = 12cm,and{\text{ }}BC = 6cm$. The angle $B$ is
A) ${120^0}$
B) ${60^0}$
C) ${90^0}$
D) ${45^0}$

Answer 1

Hint: Try to apply the Pythagoras theorem on the given sides of the triangle. Then use the converse of this theorem to deduce the type of triangle and then find the desired angle.

Complete step-by-step answer:
Given the problem, we have a triangle $ABC$ with dimensions

$

  AB = 6\sqrt 3 cm \\

  AC = 12cm \\

  BC = 6cm \\

$

We need to find angle $B$.

The Pythagoras theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other sides.

${\left( {Hypotenuse} \right)^2} = {\left( {Base} \right)^2} + {\left( {Height} \right)^2}{\text{ (1)}}$

The converse of Pythagoras theorem states that if condition $(1)$is valid for a triangle, then that triangle is a right-angle triangle.

In the given triangle $ABC$, if we consider $AC$ as the hypotenuse and $AB,BC$ as the other two sides, then

$

  A{C^2} = A{B^2} + B{C^2}\,{\text{ (2)}} \\

   \Rightarrow {\left( {12} \right)^2} = {\left( {6\sqrt 3 } \right)^2} + {\left( 6 \right)^2} \\

   \Rightarrow 144 = 108 + 36 = 144 \\

   \Rightarrow LHS = RHS \\

$

Hence equation $(1)$ is verified for $\Delta ABC$.

Since the converse of Pythagoras theorem is valid for $\Delta ABC$, this implies that $\Delta ABC$ is a right-angle triangle.

Comparing equations $(1)$and $(2)$, we get

$
  Hypotenuse = AC \\

  Base = AB \\

  Height = BC \\

$

 Hence$\Delta ABC$ is a right-angle triangle, right angled at vertex $B$.

Therefore, $\angle B = {90^0}$.
Hence option is (C). ${90^0}$ is the correct answer.

Note: It is important to note that Pythagoras theorem is only applicable on right angle triangles. Converse of Pythagoras theorem can be used to find if the given triangle is the right triangle or not. Also, Pythagoras theorem is useful in calculating the length of a side of a right triangle if the other two sides are given. It is applicable in square and rectangle to find the length of the diagonals.