Question

The sides of a triangle are 11 cm, 15 cm and 16 cm. What is the altitude to the largest side?
A.\[30\sqrt 7 \] cm
B.\[\dfrac{{15\sqrt 7 }}{2}\] cm
C.\[\dfrac{{15\sqrt 7 }}{4}\] cm
D.30 cm

Answer 1

Hint: Here, we will first find the semi perimeter of the triangle, that is, \[s = \dfrac{{a + b + c}}{2}\]. Then use the heron’s formula of area of the triangle is \[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle. Apply this formula of area of triangle, and then use the formula, \[{\text{Area}} = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\] to find the required value.

Complete step-by-step answer:
We are given the sides of a triangle are 11 cm, 15 cm and 16 cm.
We know that the semi perimeter of the circle is calculated by dividing the sum of three sides by 2, that is, \[s = \dfrac{{a + b + c}}{2}\].
We will find the values of \[a\], \[b\] and \[c\] are the sides of the triangle.
\[a = 11\]
\[b = 15\]
\[c = 16\]
We will now substitute these values in the above formula to find the semi perimeter of the given triangle.
\[
   \Rightarrow s = \dfrac{{11 + 15 + 16}}{2} \\
   \Rightarrow s = \dfrac{{42}}{2} \\
   \Rightarrow s = 21{\text{ cm}} \\
 \]
We know that the area of the triangle is calculated by\[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the three sides of the triangle.
Substituting these values of the semi perimeter of the triangle and the sides of the triangle \[a\], \[b\] and \[c\] \[h\] in the above formula of area of the triangle, we get
\[
  Area = \sqrt {21\left( {21 - 11} \right)\left( {21 - 15} \right)\left( {21 - 6} \right)} \\
   = \sqrt {21 \times 10 \times 6 \times 5} \\
   = \sqrt {6300} \\
   = 30\sqrt 7 {\text{ c}}{{\text{m}}^2} \\
 \]

Thus, the area of the triangle is \[37\sqrt 7 {\text{ c}}{{\text{m}}^2}\].

We know that the area of triangle using the formula, \[{\text{Area}} = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\], where we have base Is AC and height is \[p\].
\[ \Rightarrow {\text{Area}} = \dfrac{1}{2}\left( {AC \times p} \right)\]

Here, we know that the longest side is 16 cm, so AC is 16 cm, we get
\[
   \Rightarrow 30\sqrt 7 = \dfrac{1}{2}\left( {16 \times p} \right) \\
   \Rightarrow 30\sqrt 7 = 8p \\
 \]

Dividing the above equation by 8 on both sides, we get
\[
   \Rightarrow \dfrac{{30\sqrt 7 }}{8} = \dfrac{1}{8}\left( {8p} \right) \\
   \Rightarrow \dfrac{{15\sqrt 7 }}{4} = p \\
   \Rightarrow p = \dfrac{{15\sqrt 7 }}{4}{\text{ cm}} \\
 \]
Hence, option C is correct.

Note: In solving these types of questions, Heron’s formula should be used to compute the area of a triangle where sides of the triangle are given. Students can also find the area of triangle of using the formula, \[{\text{Area}} = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\] in the start then use the heron’s formula, but this way could take a little longer as we have to find the height of the triangle first.