# Lesson HOW TO solve problems on the triangle sides measures - Examples

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## How to solve problems on the triangle sides measures - Examples

In this lesson you will find the solutions of the typical problems that relate to the triangle sides measures.
There are two major approaches, regarding the strategy of the solution.
First, think whether the direct calculations may resolve the problem.
Alternatively, you can consider the solution using the reduction to the some kind of a linear equation.
The examples below show how each of these approaches works.
The Problems 1, 2 and 3 are solved using the direct calculations.
The Problems 4 and 5 are solved using the reduction to the linear equation.

Problem 1
Find the perimeter of the triangle with the sides of 8 cm, 5 cm and 9 cm long.

Solution
The perimeter of the triangle is the sum of the measures of its sides.
Therefore, in our case the perimeter of the triangle is equal to
8 cm + 5 cm + 9 cm = 22 cm.

Answer. The perimeter of the triangle is 22 cm.

Problem 2
Find the perimeter of the triangle, if the first side is of 8 cm long, the second side is in 4 cm longer,
and the third side measure is half of the sum of that of the first and the second sides.

Solution
The second side is 8 cm + 4 cm = 12 cm long in accordance with the problem condition.
The third side is (8 cm + 12 cm)/2 = 10 cm long in accordance with the problem condition.
The perimeter of the triangle is the sum of the measures of its sides. Therefore, it is equal to
8 cm + 12 cm + 10 cm = 30 cm.

Answer. The perimeter of the triangle is 30 cm.

Problem 3
Find the missed side of the triangle, if the first side is 7 cm long, the second side is 2 cm shorter
and the perimeter of the triangle is 16 cm.

Solution
The second side is 7 cm - 2 cm = 5 cm long in accordance with the problem condition.
Since the perimeter of the triangle is 16 cm and the two sides measures are 7 cm and 5 cm, the missed third side length is
16 cm - (7 cm + 5 cm) = 16 - 12 cm = 4 cm.

Answer. The length of the missed third side of the triangle is 4 cm.

Problem 4
In a triangle, the second side is in 7 cm longer than the first one and the third side is in 4 cm shorter than the second one.
Find the measures of the triangle sides, if the perimeter of the triangle is 40 cm.

Solution
Let x be the first side measure (in centimeters).
Then the second side is x + 7 cm, and the third side is (x + 7) - 4 = x + 3 cm in accordance with the problem condition.
Since the perimeter of the triangle is equal to 40 cm, you can write the equation
x + (x +7) + (x + 3) = 40.
Simplify this equation step by step and solve it:
x + x + 7 + x + 3 = 40,
3x + 10 = 40,
3x = 40 - 10,
3x = 30,
x = 10.

So, the first side of the triangle is 10 cm long.
Hence, the second side is 10 cm + 7 cm = 17 cm long, and the third side is 17 cm - 4 cm = 13 cm long in accordance with the problem condition.
You can check that the sum of the side measures is equal to the given value of the perimeter:
10 cm + 17 cm + 13 cm = 40 cm.

Answer. The sides of the triangle are 10 cm, 17 cm and 13 cm long.

Problem 5
Find the measures of the triangle sides, if the second side is in 6 cm shorter than the first one,
the third side measure is half of the sum of the measures of the first and the second sides, and
the perimeter of the triangle is 30 cm.

Solution
Let x be the length of the first side (in centimeters).
Then the second side is of x - 6 cm long in accordance with the problem condition,
and the third side is of (x + (x-6))/2 cm long.
Since the perimeter of the triangle is equal to 30 cm, you can write the equation
x + (x - 6) + (x +(x-6))/2 = 30.
Simplify this equation step by step and solve it:
2x + 2(x - 6) + (x +(x-6)) = 60 (after multiplying both sides of the previous equation by 2),
2x + 2x + x + x - 12 - 6 = 60,
6x - 18 = 60,
6x = 60 + 18,
6x = 78,
x = 13.

So, the first side is 13 cm long.
Hence, the second side is of 13 cm - 6 cm = 7 cm long, the third side is (13 + 7)/2 = 20/2 = 10 cm long in accordance with the problem condition.
You can check that the sum of the side measures is equal to the given value of the perimeter:
13 cm + 7 cm + 10 cm = 30 cm.

Answer. The triangle side measures are 13 cm, 7 cm and 10 cm.

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