Question

In a triangle, the sides are given as \[11cm,12\;cm\] and \[13\;cm\] .The length of the altitude corresponding to the side having length \[12\;cm\] is
A. \[10.25\;cm\]
B. \[11.25\;cm\]
C. \[12.25\;cm\]
D. \[9.25\;cm\]

Answer 1

Hint: In this problem, a triangle sides values are given in the question. So, we need to find the length of the altitude corresponding to the side. Here, we use Heron's formula to calculate the area of any given triangle by using the formula,
  \[Area = \sqrt {s(s - a)(s - b)(s - c)} \] .
Where, ‘ \[s\] ’ is the semi-perimeter of the triangle, \[s = \dfrac{{a + b + c}}{2}\] .

Complete step-by-step answer:
In the given problem,
Let a triangle ABC whose sides are a, b and c respectively.
 \[a = 11cm,b = 12cm,c = 13\;cm\]
Let the length of the altitude, \[base = 12\;cm\]
According to heron's formula, we can find the area of any given triangle provided the sides of the triangle by using the formula,
 \[Area = \sqrt {s(s - a)(s - b)(s - c)} \]
By substitute the given sides values to the formula, we get
 \[Area = \sqrt {s(s - 11)(s - 12)(s - 13)} \to (1)\]
We need to find ‘s’ values by the semi-perimeter formula, we have
 \[s = \dfrac{{a + b + c}}{2}\]
Substitute the three sides into the formula, we get
 \[s = \dfrac{{11 + 12 + 13}}{2}\]
To simplify, we get
 \[
  s = \dfrac{{36}}{2} \\
  s = 18cm \;
 \]
By applying ‘s’ value into the equation \[(1)\]
 \[(1) \Rightarrow Area = \sqrt {18(18 - 11)(18 - 12)(18 - 13)} \]
By simplify in further, we get
 \[Area = \sqrt {18(7)(6)(5)} = \sqrt {3180} \]
Therefore, the area is \[61.5c{m^2}\]
Now, we have the base value is \[12\;cm\]
Area of a triangle is \[\dfrac{1}{2}base \times height\]
By substitute the base value in further step, we get
Area of a triangle \[ = \dfrac{1}{2} \times 12 \times h\]
To simplify the multiplication, we get
Area of a triangle \[ = 6h\]
Where, the area is \[61.5c{m^2}\] .
By substitute the value, we get
 \[
  h = \dfrac{{61.5}}{6} \\
  h = 10.25cm \;
 \]
Therefore, the final answer is option (A) \[h = 10.25\;cm\] .
So, the correct answer is “OPTION A”.

Note: In this question, we have all the three sides of the triangle a,b,c are given. So we need to find the length of the altitude by using heron's formula. Thus, the triangle can be given by \[Area = \sqrt {s(s - a)(s - b)(s - c)} \] .
Where, ‘ \[s\] ’ is the semi-perimeter of the triangle, \[s = \dfrac{{a + b + c}}{2}\] .we have to remember this formula to calculate this type of question.