The perimeter of a triangle is 27 cm. What are the lengths of the sides of a triangle if its second side is 2 cm longer than the first, and the third side is 4 cm shorter than the second?
Consider the triangle given in the problem statement. This triangle:
denote the lengths of the sides as a, b, and c, respectively;
the perimeter of this triangle is 27 cm (respectively).
We know that the perimeter of any arbitrary figure is found as the sum of the lengths of all its sides. In our case, the figure is a triangle and has three sides with the appropriate designations. Therefore, based on the above, we can write that the perimeter of the triangle is based on the formula:
P = a + b + c
From the condition of the problem, we know that the perimeter is 27 cm.Therefore, the formula for sha will take the following form:
a + b + c = 27 (1)
Find the lengths of the sides of this triangle
Suppose the first side is the side with the longest a.
Then, from the condition of the problem, the second side (with the longest b) is 2 cm longer than the first. That is, we can write this condition as:
b = a + 2 (2)
We also know that the third side (with length c) is 4 cm shorter than the second side. Therefore, we will write it as:
c = b – 4 (3)
Substitute expression (2) into formula (3) and get the following:
c = b – 4 = a + 2 – 4 = a – 2 (4)
Let’s return to formula (1) and substitute the resulting values from (2) and (4) into it. We get:
a + b + c = a + a + 2 + a – 2 = a + a + a + 2 – 2 = a * (1 + 1 + 1) = a * 3;
a * 3 = 27;
a = 27/3;
a = 9 cm
We now know the length of the first side. Based on this, we can find the lengths of the remaining sides:
b = a + 2 = 9 + 2 = 11 cm;
c = a – 2 = 9 – 2 = 7 cm.
Answer: 9 cm, 11 cm, 7 cm