Question

The sides of a triangle are 16cm, 12cm and 20cm . Find the height of the triangle, corresponding to the smallest side.

Answer 1

Hint: Use heron’s formula to find the area of the given triangle then compare that area with the regular area formula of a triangle that is \[\dfrac{1}{2} \times b \times h\] from here you can easily get the value of height.

Complete Step by Step solution:
First of all let us try to draw the figure of the triangle described in the question.

Here \[a = 16cm,b = 12cm\& c = 20cm\]
Now let us write the Heron’s Formula first
\[A = \sqrt {s(s - a)(s - b)(s - c)} \]
Here a,b,c are the lengths of the sides of the triangle and s is the semi-perimeter which is given by the formula \[s = \dfrac{{a + b + c}}{2}\]
So let us try to find the value of s i.e.,
\[\begin{array}{l}
\therefore s = \dfrac{{a + b + c}}{2}\\
 \Rightarrow s = \dfrac{{16 + 12 + 20}}{2}\\
 \Rightarrow s = \dfrac{{48}}{2}\\
 \Rightarrow s = 24
\end{array}\]
Now we have all the values let's put it in the formula to find the area of triangle
\[\begin{array}{l}
\therefore A = \sqrt {s(s - a)(s - b)(s - c)} \\
 \Rightarrow A = \sqrt {24(24 - 16)(24 - 12)(24 - 20)} \\
 \Rightarrow A = \sqrt {24 \times 8 \times 12 \times 4} \\
 \Rightarrow A = \sqrt {96 \times 96} \\
 \Rightarrow A = 96c{m^2}
\end{array}\]
Now we know that area of the triangle is also given by
\[A = \dfrac{1}{2} \times b \times h\]
Therefore if we equate them we will get it as
\[\begin{array}{l}
 \Rightarrow 96 = \dfrac{1}{2} \times b \times h\\
 \Rightarrow b \times h = 192
\end{array}\]
Now it was mentioned in the question to find the height corresponding to the smallest side, i.e. 12 so in place of b in the formula we will put 12. So we get it as
\[\begin{array}{l}
 \Rightarrow 12 \times h = 192\\
 \Rightarrow h = \dfrac{{192}}{{12}}\\
 \Rightarrow h = 16cm
\end{array}\]
Therefore the height of the triangle corresponding to the smallest side is 16cm.

Note: Be careful while putting values in the heron's formula. Interchanging the values could lead to a negative sign in the answer and if you encounter such a situation then check the solution correctly. Also give the units wherever necessary that can also result in deduction of marks in examinations.