Question

The lengths of the sides of a triangle are in the ratio \[3:4:5\] and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side

Answer 1

Hint:
Here, we will use the formula of perimeter and find each side of the triangle. Then using Heron’s formula, we will find the required area of the triangle. We will then equate it to the general formula of area of a triangle by taking base as the longest side. Solving this, we will be able to find the required height corresponding to the longest side.

Formulas Used:
We will use the following formulas:
1) According to Heron’s formula, Area of triangle\[ = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where, \[s\] is the semi-perimeter, and \[a,b,c\] are the three sides of the triangle.
2) Area of a triangle \[ = \dfrac{1}{2} \times {\rm{base}} \times {\rm{height}}\]

Complete step by step solution:
We know that the lengths of the sides of a triangle are in the ratio \[3:4:5\].
Hence, let the three sides of the triangle be \[3x\], \[4x\] and \[5x\] respectively.
Now, it is given that the perimeter of the triangle is 144 cm.
Therefore, the sum of all the three sides of the triangle is 144 cm.
\[ \Rightarrow 3x + 4x + 5x = 144\]
\[ \Rightarrow 12x = 144\]
Dividing both sides by 12, we get,
\[ \Rightarrow x = 12\]
We will now substitute the value \[x\] in \[3x\], \[4x\] and \[5x\] to find the sides of the triangle.
Therefore, the sides of the triangle are:
\[3x = 3 \times 12 = 36{\rm{cm}}\]
\[4x = 4 \times 12 = 48{\rm{cm}}\]
\[5x = 5 \times 12 = 60{\rm{cm}}\]
Now, the semi-perimeter of the triangle,\[s = \dfrac{{144}}{2} = 72\]
We will use Heron’s formula to find the area of the triangle because all the three sides of the triangle are known.
Substituting \[s = 72\], \[a = 36\], \[b = 48\] and \[c = 60\] in the formula Area of triangle\[ = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \],we get,
Area of triangle\[ = \sqrt {72\left( {72 - 36} \right)\left( {72 - 48} \right)\left( {72 - 60} \right)} \]
Subtracting the terms inside the bracket, we get
\[ \Rightarrow \]Area of triangle \[ = \sqrt {72\left( {36} \right)\left( {24} \right)\left( {12} \right)} \]
Simplifying the expression, we get
\[ \Rightarrow \]Area of triangle \[ = 288 \times 3 = 864\]
Therefore, Area of the given triangle\[ = 864{\rm{c}}{{\rm{m}}^2}\]
Now, as we can clearly see, the longest side of this triangle is of the length 60 cm.
Substitute the area of triangle\[ = 864{\rm{c}}{{\rm{m}}^2}\], the base as 60 cm and height as \[h\], in the formula Area of a triangle \[ = \dfrac{1}{2} \times {\rm{base}} \times {\rm{height}}\], we get
\[ \Rightarrow 864 = \dfrac{1}{2} \times 60 \times h\]
\[ \Rightarrow 864 = 30 \times h\]
Dividing both sides by 30, we get
\[ \Rightarrow \dfrac{{864}}{{3 \times 10}} = h\]
\[ \Rightarrow h = \dfrac{{288}}{{10}}\]
Converting this as decimal,
\[ \Rightarrow h = 28.8{\rm{cm}}\]

Therefore, the area of the triangle is 864 square centimetres and the height corresponding to the longest side is 28.8 cm.
Hence, this is the required answer.

Note:
A triangle is a two-dimensional geometric shape that has three sides. There are different types of triangles such as equilateral triangle, right-angled triangle, isosceles triangle etc. We have used Heron’s formula to find the sides of the triangle and not the Pythagoras theorem. This is because Pythagoras theorem can only be used to find the sides of a right-angled triangle, whereas we can find sides of any triangle using Heron’s formula.