Question

In triangle ABC, angle B is the right angle. If a =16 and c = 12, then b = ?
a. 8
b. 18
c. 20
d. 28

Answer 1

Hint: We will be using the cosine rule for the solution of this question.
The cosine rule is used when we are given either:-
a) three sides
b) two sides and the included angle.

For the solution of this question, we will be using the formula:
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-2ac\cos B\]

Complete step-by-step answer:

We have been given the two sides a = 16 and c = 12 and we have also been given the included angle which the two sides ‘a’ and ‘c’ contain between them.
Using the formula provided in the hint, we are going to substitute the values of a and c in the formula as shown here:
\[{{b}^{2}}={{\left( 16 \right)}^{2}}+{{\left( 12 \right)}^{2}}2\times 16\times 12\times \cos {{90}^{\circ }}\]
We know that the value of \[\cos {{90}^{\circ }}\] is 0, hence the formula can be simplified as:
\[\begin{array}{*{35}{l}}
   {{b}^{2}}=\text{ }256\text{ }+\text{ }144\text{ }\text{ }2\times 16\times 12\times 0 \\
   \begin{align}
  & =400\text{ } \\
 & ={{\left( 20 \right)}^{2}} \\
\end{align} \\
\end{array}\]
Now, on taking square root on both the side, we get
Therefore, b = 20.
Hence, the answer is (c) 20.

Note: There is another rule, called the sine rule. The sine rule is used when we are given either:-
a) two angles and one side
b) two sides and a non-included angle.
If the student uses the sine rule by mistake, then the answer that he will get will be wrong.