Question

Find the value of c in a triangle, if the semi perimeter is $13$ cm and the values of a and b are $8$ cm and $7$ cm respectively.

Answer 1

Hint : Perimeter is the sum of all the sides of a triangle. The semi perimeter is half of the sum of all the sides of a triangle i.e. semi perimeter = perimeter of the triangle/ 2. In this question, we will first assume its semi perimeter. And then, substituting the values in the formula, we can find the value of c which is the side of the triangle. Further you can Heron's formula to find the area of the triangle (if asked).

Complete step-by-step answer :
Let, S be the semi perimeter of the triangle.
Given that, the semi perimeter of the triangle is S = $13$ cm.
Given that, the three sides of the triangle are a, b and c.
The values of a = $8$ cm and b = $7$ cm.
We need to find the value of c.
We know that,
The perimeter (P) of the triangle is twice the semi perimeter (S).
$\therefore P = 2S$
Substituting the value of S, we will get,
$\Rightarrow P = 2(13)$
$\Rightarrow P = 26$ cm
Also, we know that,
The perimeter of the triangle is the sum of the three sides.
$\therefore P = a + b + c$
Substituting the values in the above equation, we will get,
$\Rightarrow 26 = 8 + 7 + c$
$\Rightarrow 26 = 15 + c$
By using transposition, and moving the RHS term to LHS, we will get,
$\Rightarrow 26 - 15 = c$
$\Rightarrow 11 = c$
Rearrange this equation, we will get,
$\Rightarrow c = 11$ cm

Another Method:
We will use Heron's formula.
Thus, the semi perimeter of the triangle is
$\therefore S = \dfrac{{a + b + c}}{2}$
Substituting the values in the above equation, we will get,
$\Rightarrow 13 = \dfrac{{8 + 7 + c}}{2}$
$\Rightarrow 13(2) = 15 + c$
$\Rightarrow 26 = 15 + c$
$\Rightarrow 11 = c$
Rearrange this equation, we will get,
$\Rightarrow c = 11$ cm
Hence, the value of c in a triangle is $11$ cm when the values of a and b are $8$ cm and $7$ cm respectively and the semi perimeter is $13$ cm.
So, the correct answer is “$13$ cm”.

Note : Heron's formula was first given by Heron of Alexandria. It is used to find the area of different types of triangles like equilateral, isosceles, and scalene triangles or quadrilaterals. Heron's formula is used to find the area of triangles when lengths of all their sides are given or the area of quadrilaterals. It is also known as Heron's formula. The use of this formula for finding the area does not depend on the angles of a triangle. Its formula is: Area = $\sqrt {s(s - a)(s - b)(s - c)}$ where s is the semi perimeter and a, b and c are sides of the triangle.