Question

The perimeter of a triangle whose sides are in A.P. is 21 cm and the product of lengths of the shortest side and the longest side exceeds the length of the other side by 6 cm. The longest side of the triangle is
A. 1 cm
B. 7 cm
C. 13 cm
D. None

Answer 1

Hint: The perimeter of a triangle, \[P = a + b + c\] units is known from the definition of the perimeter. Here, "a," "b," and "c" are the triangle's sides. We will enter the length of the triangle's sides into the above formula and solve it to get the perimeter expression.

Formula Used: Triangle’s perimeter can be determined by
\[{\rm{Perimeter = a + b + c units}}\]

Complete step by step solution: We have been provided in the question that,
The triangle's perimeter whose sides are in Arithmetic progression is \[21\] cm and the product of lengths of the shortest side and the longest side exceeds the length of the other side by \[6\] cm
Now, let us consider that the sides of the triangles be
\[a - d,{\rm{ }}a,{\rm{ }}a + d\]
We have been provided that the perimeter of a triangle is
\[ = 21\] cm
Now, we can write it as,
\[a - d + a + a + d = 21\]
Now, we have to group the like terms to simplify, we have
\[3a{\rm{ }} = {\rm{ }}21\]
Now, we have to solve for a, for that we have to move 3 to the right hand side of the equation, we get
\[a = \dfrac{{21}}{3}\]
The above equation have to be further simplified, we will obtain
\[a = 7\]
The product of the dimensions of the shortest and longest sides is 6 cm longer than the length of the opposite side.
Therefore, we have
\[\left( {a - d} \right)\left( {a + d} \right){\rm{ }}-{\rm{ }}a{\rm{ }} = {\rm{ }}6\]
Now, we have to multiply the two terms \[\left( {a - d} \right)\]\[\left( {a + d} \right)\] we get
\[{a^2}\;-{\rm{ }}{d^2}\;-{\rm{ }}a{\rm{ }} = {\rm{ }}6\]
Now, on substituting the values, we have
\[{7^2}\;-{\rm{ }}{d^2}\;-{\rm{ }}7{\rm{ }} = {\rm{ }}6\]
On solving the powers of the above equation, we obtain
\[42-6 = {d^2}\]
Now, let’s subtract the number \[42-6\] we get
\[{d^2}\; = {\rm{ }}36\]
On both sides, we have to take square roots we will get
\[d{\rm{ }} = {\rm{ }}6\]
Now, we can conclude that the sides of the triangles are
\[1,7\& 13\]
Therefore, the longest side of the triangle is \[13\] cm

Option ‘C’ is correct

Note: We have applied the formula to calculate the perimeter. The perimeter will be measured in the same unit as the lengths of a side or triangle. The unit of the area will be the square of the unit of length of a triangle. So, one should not neglect to include the unit. Students should be very cautious in solving these types of problems.