Question

The semi-perimeter of a triangle is \[96cm\] and its sides are in the ratio \[3:4:5\]. Find the area of the triangle.

Answer 1

Hint: First, assume \[t\] as an unknown numeric value. Use Herson's perimeter formula to find the sides of the triangle and Heron's area formula to find the area of the triangle. Apply prime factorization method to find the square root for the value. Multiplying the obtained factors we get the required area of the triangle.
Formula used:
Area of triangle \[ = \sqrt {S(S - a)(S - b)(S - c)} \]
\[S = \dfrac{{a + b + c}}{2}\]

Complete step by step answer:
Let \[t\] be some integer value. sides are in the ratio \[3:4:5\] .
Now include the ratio value to \[t\] .
Consider the sides assuming that \[3t,4t,5t\] .
Used Heron's formula,
\[A = \sqrt {S(S - a)(S - b)(S - c)} \]
\[A = \sqrt {S(S - a)(S - b)(S - c)} \]
\[S\] denoted by a semi-perimeter of the triangle.
Semi-perimeter of triangle \[ = \dfrac{{\left( {sum{\text{ }}of{\text{ }}the{\text{ }}sides{\text{ }}of{\text{ }}triangle} \right)}}{2}\]
Sum of the sides of the triangle \[ = a + b + c\]
Here \[a = 3t,b = 4t,c = 5t\]
\[96 = \dfrac{{3t + 4t + 5t}}{2}\]
\[192 = 12t\]
\[t = \dfrac{{192}}{{12}}\]
\[t = 16\]
So that now substitute \[t\] value in \[a,b\] and \[c\]
\[a = 3(16)\]
\[ = 48cm\]
\[b = 4(16)\]
\[ = 64cm\]
\[c = 5(16)\]
\[ = 80cm\]
Now apply \[S,a,b\] and \[c\] values in Heron's area formula
Area of triangle \[ = \sqrt {S(S - a)(S - b)(S - c)} \]
Area of triangle \[ = \sqrt {96(96 - 48)(96 - 64)(96 - 80)} \]
\[ = \sqrt {96(48)(32)(16)} \]
Now used prime factorization method,
The prime factorization method means writing the number as the product of its prime factors.
Find the prime factors of \[96,\,48,\,32,\,16\]
Prime factors of,
\[96 = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 3} \]
\[48 = \sqrt {2 \times 2 \times 2 \times 2 \times 3} \]
\[32 = \sqrt {2 \times 2 \times 2 \times 2 \times 2} \]
\[16 = \sqrt {2 \times 2 \times 2 \times 2} \]
\[ = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} \]
Take square root,
\[ = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\]
Product the values,
\[ = 1536\]
This is finding the area sum so we included the \[c{m^2}\] in the final answer.
So that area is \[1536c{m^2}\].

Note:
Assuming \[t\] as a numeric constant and finding its value using Herson’s perimeter formula helps to find the sides of the triangle and the method of prime factorization helps the numbers to get its prime factors. In the final answer, we include the unit as \[c{m^2}\]because in finding an area units are mandatory. In case of volume, we include \[c{m^3}\] as its unit.