Question

In the given figure $\angle ABC={{90}^{\circ }}$ and $BD\bot AC$. If AB = 5.7cm, BD = 3.8cm and CD = 4.4cm then find the length of side BC.

Answer 1

Hint: Now in the given triangle BD is perpendicular to AC hence we can say that $\angle CDB={{90}^{\circ }}$ . Hence, we get that CBD is a right angle triangle. Hence we can use Pythagoras theorem in the triangle. Now according to Pythagoras theorem we get \[B{{D}^{2}}+D{{B}^{2}}=B{{C}^{2}}\] . Hence we will substitute the values of BD and DB in the equation to find $B{{C}^{2}}$ . Now taking the square root we will get BC.

Complete step by step answer:
Now consider the given triangle

Now in triangle ABC we are given that BD is perpendicular to AC. Hence we can say $\angle CDB={{90}^{\circ }}$ .
Now let us just consider the triangle CDB.
Since we have $\angle CDB={{90}^{\circ }}$ we can say triangle CBD is a right angle triangle.
Now in triangle CBD we know that $\angle CDB={{90}^{\circ }}$ , $CD=4.4,BD=3.8$and BC is the hypotenuse.
Hence we can use Pythagoras theorem
According to Pythagoras theorem the sum of squares of perpendicular sides is equal to square of hypotenuse
Here we have BD and DB are adjacent sides and BC is the hypotenuse. Hence we get
\[\begin{align}
  & B{{D}^{2}}+D{{B}^{2}}=B{{C}^{2}} \\
 & \Rightarrow {{\left( 3.8 \right)}^{2}}+{{\left( 4.4 \right)}^{2}}=B{{C}^{2}} \\
 & \Rightarrow 14.44+19.36=B{{C}^{2}} \\
 & \therefore B{{C}^{2}}=33.80 \\
\end{align}\]

Now taking square root on both sides we get $BC=\sqrt{33.80}$

Note: We can also solve this question with an alternate method. Now first we will consider triangle ABD we know that $BD\bot AC$. Hence $\angle ADB={{90}^{\circ }}$. Now we will use Pythagoras theorem to find AD. Now we know that AD + DC = AC. Hence we will find AC. Now we have $\angle ABC={{90}^{\circ }}$ hence we will use Pythagoras theorem to find BC.