Question

The lengths of the sides of a triangle are in the ratio \[5:6:7\]. What is the length of the longest side if the perimeter is less than \[54{\text{ }}cm\]?

Answer 1

Hint: Here, the ratio of the lengths of the sides of a triangle is given as \[5:6:7\]. We will assume the common ratio as \[x\] to get the sides as \[5x\], \[6x\] and \[7x\]. Then, the perimeter of a triangle is the sum of all the three sides of the triangle and given that perimeter is less than \[54{\text{ }}cm\], so we will write \[5x + 6x + 7x < 54\].Then we will simplify it to find the length of the longest side.

Complete step by step answer:
Given, the lengths of the sides of a triangle are in the ratio \[5:6:7\]. Let the common ratio be \[x\]. So, the lengths of the sides of a triangle will become \[5x\], \[6x\] and \[7x\].

Given, the perimeter is less than \[54{\text{ }}cm\].
As we know, the perimeter of a triangle is the sum of all the three sides of the triangle. So, we can write
\[ \Rightarrow 5x + 6x + 7x < 54{\text{ }}\]
On simplifying, we get
\[ \Rightarrow 18x < 54{\text{ }}\]

As we know that the inequality remains the same if we divide it by a positive number.
On dividing both the sides by \[18\], we get
\[ \Rightarrow x < 3{\text{ }}\]
So, the lengths of the sides of a triangle will become less than \[\left( {5 \times 3} \right)\], \[\left( {6 \times 3} \right)\] and \[\left( {7 \times 3} \right)\] i.e., \[{\text{15 }}cm\], \[{\text{18 }}cm\] and \[{\text{21 }}cm\].

Therefore, the length of the longest side is less than \[{\text{21 }}cm\].

Note: Perimeter is a path or the boundary that surrounds a shape and can also be defined as the length of the outline of a shape. Depending upon the dimension, the perimeter of different figures can be equal in measure. Also note that the perimeter of a circle is called its circumference and its formula is \[{\text{2}}\pi \times \left( {radius} \right)\].